TESTS  OF  THE  EFFECT  OF  BRACKETS 

IN 

REINFORCED  CONCRETE  RIGID  FRAMES 


BY 


F^IANK  ERWIN  RICHART 

B.S.  in  C.E.  University  of  Illinois,  1914 
M.S.  in  C.E.  University  of  Illinois,  1915 


THESIS 

SUBMITTED  IN  PARTIAL  FULFILLMENT  OP  THE  REQUIREMENTS 
■ FOR  THE  DEGREE  OP  CIVIL  ENGINEER 
IN  THE  GRADUATE  SCHOOL  OP  THE  UNIVERSITY 
OF  ILLINOIS,  1922 


URBANA,  ILLINOIS 


Digitized  by  the  Internet  Archive 
in  2015 


https://archive.org/details/testsofeffectofbOOrich 


Ou. 


UNIVERSITY  OF  ILLINOIS 

THE  GRADUATE  SCHOOL 

April  4^ 192^ 


I HEREBY  RECOMMEND  THAT  THE  THESIS  PREPARED  BY. 


VRAMK  ERWTW  RICHART  . 


ENTITLED  TESTS  OF  THE  EFFECT  OE  BRACKETS  IN  RETTJ FjQBCEL. 


CONCRETE  RIGID  FRAMES 


BE  ACCEPTED  AS  FULFILLING  THIS  PART  OF  THE  REQUIREMENTS  FOR  THE 


PROFESSIONAL  DEGREE  OF 


Head  of  Department  of  CIVTI.  ENCTNEERTNC, 


Recommendation  concurred  in: 





Committee 


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CONTENTS 


I.  INTRODUCTION.  Page 

1.  Preliminary 1 

2,  Nature  of  Investigation. i 2 

II.  ANALYTICAL  TREATMENT. 

3*  Analysis  of  the  Effect  of  Brackets  4 

4.  Effect  of  Brackets  with  Different  Loadings  ...  .19 

5.  Effect  of  Brackets  in  Various  Tjrpes  of  Frames.  . .21 

III.  TEST  SPECIMENS  AND  APPARATUS. 

6.  Description  of  Test  Specimens 29 

7.  Materials  and  Making  of  Specimens.  . 41 

8.  Testing  Apparatus 44 

IV.  TEST  DATA  AND  DISCUSSION  OP  RESULTS. 

9.  Procedure  and  Phenomena  of  Tests  48 

10.  Thrusts  and  Moments 60 

11.  Flexural  Stresses  and  Deformations 64 

12.  Shearing  Stresses. .74 

13.  Moment  of  Inertia. 77 

14.  Deflections 85 

15.  Conclusions 91 

V.  APPENDIX  I. 

16.  Test  Data  and  Drawings 98 

VI.  APPENDIX  II. 

17.  Supplementary  Tests  of  Paper  Models. 131 


-II- 


±;IST  OF  PIGUIilS 

Wo. 

Page 

1.  Outline  Drawing:  Used,  in  Analysis  of  i^'rame 

2»  Eelations  between  Moments  and  Eeactions*  ••••••••  7 

3.  Eelations  between  Calculated  Moment  and  i^lze  of  Bracket.  9 

4.  Calculated  Effect  of  Bracket  with  Varying  Proportions 

of  Frame 

5.  Eelation  between  Slenderness  of  Frame  and  Calculated 

Eeduction  in  Moment  Due  to  Bracket.  

6.  Effect  of  Length  of  Clear  Span  upon  Moment  at  Midspan.  • 14 

7.  Calculated  Moments  in  Frames  of  Varying  Moments  of  Inertia.  18 

8.  Influence  Lines  for  Horizontal  Eeactions  of  Frames..  . . EO 

9.  Moments  at  Midspan  of  Frames  of  Different  Types  and 

Proportions 

10.  iJistanoe  hj,  Contraflexnre.  25 

11.  Moments  at  Midspan  of  Continuous  Beams  with  Brackets.  . E7 

12.  Details  of  Test  Specimen  13A1.  .,,*50 

13.  Details  of  Test  Specimen  13A2.  

14.  Details  of  Test  Specimen  13B1 ,52 

15.  Details  of  Test  Specimen  13B2 

16.  Details  of  Test  Specimen  13C1 

17.  Details  of  Test  Specimen  13C2 ..55 

18.  Details  of  Test  Specimen  13D1 

19.  Details  of  Test  Specimen  13D2 

20.  Details  of  Test  Specimen  13E1 

21.  General  view  of  Test  Apparatus.  •••........,45 

22.  General  Arrangement  of  Test  Apparatus 

23.  Views  of  Specimen  13A1  and  13aE  after  Test 50 


-III- 


9 t ^ r A A * 


5 


1 

1 

1 

\ 

\ 


LIST  OF  FIGUi'OilS  -Continued. 

Lo.  Page 

24.  Views  of  Specimen  13B1  and  13B2  after  Test. 51 

25.  Viev/s  of  Specimen  13C1  and  1302  after  Test 52 

26.  views  of  Specimen  13L1  and  13D2  after  Test 53 

27.  Views  of  Specimen  1302,  13D2,  and  13B1  after  Test 54 

28.  Calculated  and  Observed  Moments  at  Midspan 61 

29.  Relative  Moments  in  Frames . . . 1^- 

30.  Observed  Stresses  in  Specimen  13A1 65 

31.  Obsenred  Stresses  in  Specimen  13a2 65 

32.  Observed  Stresses  in  Specimen  13B1 66 

33.  Observed  Stresses  in  Specimen  13B2 66 

34.  Observed  Stresses  in  Specimen  1301 67 

35.  Observed  Stresses  in  Specimen  1302 67 

36.  Observed  Stresses  in  Specimen  13L1 68 

37.  Observed  Stresses  in  Specimen  13L2.  . 68 

38.  Observed  Stresses  in  Specimen  13iill 69 

39.  Relative  stress  Distribution  in  Frames 71 

40.  L'eutral  Axes  in  Curved  Beams 73 

41.  Shear  Diagrams  for  Frames  at  Maximum  Load 75 

42.  Member  under  Combined  ilexure  and  Direct  Stress 79 

43.  Variation  in  I with  COmpressive  Stress 81 

44.  '/ariation  in  I with  Depth  of  Member 83 

45.  Calculated  and  Observed  Deflections,  Specimen  13A1 87 

46.  Calculated  and  Observed  Deflections,  Specimen  13B1 87 

47.  Calculated  and  Observed  Deflections,  Specimen  13C1 88 

48.  Calculated  and  Observed  Deflections,  Specimen  13D1.  ....  88 

-IV- 


LIST  0?  JIG-UILIS- Continued 

uo • Page 

49.  Calculated  and  Observed  Deflections , Specimen  ISEl.  ...  89 

50.  Diagram  Showing  Decrease  in  Stiffness  with  Increasing 

Load  on  Prames 92 

51-68.  Position  of  cage  Lines  and  Cracks 113 

69.  Arrangement  of  Apparatus  for  Testing  Paper  Models.  . . . 135 

70.  Values  of  H/f  from  Tests  of  Paper  Models 136 

71.  Values  of  H/p  from  Calculations,  using  Aquation  5.  . . .137 

72.  Values  of  H/p  from  Calculations  and  Tests  of  Type  A Modell38 

73.  values  of  H/p  from  Calculations  and  Tests  of  Type  D ModelJ.39 

74. .Values  of  H/p  from  Calculations  and  Tests  of  Type  C Model.140 

75.  Values  of  H/p  from  Calculations  and  Tests  of  Type  D Model.141 

76.  influence  Lines  for  Horizontal  reactions 142 

LIST  OP  TABLES 

Ho.  Page 

1.  Percentage  of  longitudinal  Beinf orcement 40 

2.  rhysical  properties  of  Steel  Bars 41 

3.  Compressive  Strength  of  Concrete .43 

4.  Data  of  Tests' 59 

5.  Comparison  of  Observed  and  Calculated  Moments 62 

6.  Comparison  of  Observed  and  Calculated  Tensile 

Stresses  in  Beinf  orcement 70 

7.  Maximum  Observed  Shearing  Stresses  in  Prames .76 

8.  Calculated  and  Observed  Deflections  at  Midspan 90 

9.  Data  of  Tests 98 

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1. 

TESTS  OP  EFFECT  ^ BRACKETS 
IN 

REINFORCBB  CONCRETE  RIGID  FRAIIES, 

INTRODUCTION. 

1.  Preliminary*-  This  thesis  is  based  upon  the  results  of 
one  of  a number  of  investigations  conducted  in  1918  by  the  Concrete 
Ship  Section  of  the  Emergency  Fleet  Corporation  in  developing 
the  design  and  construction  of  reinforced  concrete  ships*  The 
immediate  object  of  the  investigation  described  herein  was  to 
determine  the  effect  produced  upon  the  distribution  of  bending 
moments  in  a transverse  ship  frame  by  using  brackets  or  haunches 
at  the  inside  corners  of  the  frame*  The  results  of  the  investiga- 
tion have  a wider  significance,  however,  as  a contribution  to  presait 
knowledge  of  the  behavior  of  framed  structures  under  stress,  and 
their  greatest  value  undoubtedly  lies  in  their  application  to  the 
general  field  of  reinforced  concrete  construction* 

In  recent  years  considerable  attention  h^s  been  paid  to 
methods  of  analysis  of  rigidly  connected  frames  and  this  theoreti- 
cal treatment  has  been  supplemented  by  a comparatively  few  tests 
of  reinforced  concrete  bents*  However,  there  is  little  informa- 
tion available  as  to  the  correct  analysis  of  frames  with  haunches 
or  brackets  at  the  intersections  of  members  and  apparently  no  test 
data  bearing  upon  this  form  of  construction*  A v/ell-known  method 
of  analysis  has  been  used  here  and  although  its  application 
requires  a number  of  assumptions  it  seems  to  give  satisfactory 
results* 


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The  value  of  definite  knmvledge  regarding  the  effect  of 
■brackets  on  stress  distri'bution  follows  from  the  fact  that  the 
bending  moment  at  any  section  may  be  made  to  considerably 

through  their  use  and  that  the  material  used  in  brackets  is 
placed  where  it  can  resist  the  high  local  stresses  which  the 
brackets  attract  to  the  corners  of  the  frame,  Praperly  designed 
brackets  will  produce  a considerable  economy  of  weight. 

This  investigation  was  made  at  the  John  Fritz  Civil  Engi- 
neering Laboratory,  Lehigh  University,  August  to  October, 1918, 
by  the  Concrete  Ship  Section  of  the  Emergency  Fleet  Corporation 
under  the  direction  of  W,  A.  Slater, Engineer  Physicist .United 
States  Bureau  of  Standards.  Alter  the  armistice  the  data  of  the 
tests  were  taken  over  by  the  Bureau  of  Standards  and  it  is 
through  cooperation  with  tho-t  Bureau  that  the  data  have  been  made 
available  for  this  thesis.  It  is  expected  th^.t  the  Bureau  of  ^ 
St  mdards  will  publish  a Technologic  Paper  containing  the  sub;|ect 
matter  of  this  thesis.  The  opportunity  of  carrying  out  as  ex- 
tensive an  investigation  as  is  here  reported  is  due  largely  to 
the  support  of  R.  J.  7/ig,  Head  of  the  Concrete  Ship  Section,  It 
is  desired  also  to  express  full  appreciation  of  the  assistance 
afforded  by  Professor  ¥/.  P.  McZibben  and  other  officials  of 
Lehigh  University  in  extending  to  the  Concrete  Ship  Section  the 
use  of  the  laboratory  and  equipment  for  this  work.  The  investi- 
gation was  planned  and  carried  out  by  the  writer.  Valuable 
suggestions  in  laying  out  the  tests  were  received  from  C.  A, 
Maney,  Designing  Engineer  of  the  Concrete  Ship  Section,  Major 
W.  M.  Wilson  and  Major  A.  R.  Lord,  successively  in  charge  of  the 
laboratory  at  Lehigh  University  for  the  Emergency  Fleet  Corpora- 
tion, are  entitled  to  acknowledgment  for  cooperation  in  carrying 


out  the  tests, 

2,  Nature  of  Investip:ation,~An  important  use  of  brackets  in 
ship  construction  is  at  the  corners  of  transverse  frames.  Since 
a sharp  corner  at  the  intersection  of  horizontal  and  vertical 
members  produces  a section  of  highly  concentrated  fibre  stress 
combined  with  a large  negative  bending  moment,  a bracket  is  used 
primarily  to  reduce  the  compressive  stress  at  this  place.  In 
addition,  since  in  all  t3rpes  of  statically  indeterminate  frames 
the  distribution  of  bending  moments  depends  upon  the  relative 
stiffness  of  the  members  or  parts  of  which  the  structure  is  compos- 
ed, the  bracket  produces  two  other  effects; (1)  it  affects  the  dis- 
tribution of  moments  throughout  the  frame  because  of  the  local 
variation  in  stiffness  which  it  produces,  and  (2)  it  affects  the 
magnitude  of  the  negative  moment  as  well  as  the  distribution 
of  moments  because  it  changes  the  shape  of  the  axis  of  the  members 
at  the  corners  of  the  frame.  This^change  in  shape  is  sometimes 
regarded  as  a shortening  of  the  span  of  the  members  in  which  the 
brackets  are  used. 

In  choosing  the  type  of  specimen  for  these  tests,  the  object 
was  to  secure  a frame  similar  in  form  and  loading  to  a ship  fr^me, 
and  of  fairly  large  size.  A rectangular  two-legged  bent  was 
chosen  as  approximating,  in  the  inverted  position, the  lower  part  of 
a transverse  ship  frame.  The  columns  v/ere  made  hinged  at  the  bases 
in  order  to  simplify  the  interpretation  of  the  results.  The 

brackets  used  were  similar  to  those  used  in  ship  design. 

In  testing, all  specimens  were  loaded  to  failure  and  observa- 
tions of  deformation  a,nd  deflection  were  made  at  regular  increments 

of  lOw-d,  The  experimental  data  have  been  compared  with  analytical 
deductions  and  in  general  a satisfactoiy  agreement  has  been  found. 


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II.  MALYTICAL  TREAT5;TMT 

3.  Analysis  of  the  Effect  of  Brackets. - Mathematical 
analyses  of  rigidly  connected  frames  are  usually  based  upon  the 
assumption  that  the  members  of  a frame  are  uniform  in  section 
throughout  their  length,  and  there  is  little  information  on  the 
proper  way  of  analyzing  frames  in  which  abrupt  changes  in  cross 
section  occurs*.  However,  in  the  analysis  which  follows  it  was 
considered  sufficiently  accurate  for  frames  containing  brackets 
to  draw  in  the  approximate  axis  of  the  frame,  and  to  consider  as 
fully  effective  the  depth  of  the  cross-section  measured  normal  to 
this  axis.  A semi -graphical  method  often  used  in  arch  analysis 
was  then  applied  to  the  frame. 

In  the  analysis  of  the  two  hinged  frame  of  Eig.l,  which 
shows  one  of  the  types  that  was  tested,  the  outline  of  the  frame 
was  first  drawn  to  scale  and  the  axis  divided  into  a nimiber  of 
sections  approximately  equal  in  length.  The  length  of  a section 
was  denoted  by  the  depth  of  cross-section  by  d,  and  the  verti- 
cal distance  of  the  centroid  of  the  section  above  the  hinges  by 
2;.  Values  of  d and  y;  were  scaled  from  the  drawing  where 
necessary  for  each  section  of  the  frame.  Values  of  d were  used  to 
calculate  the  moment  of  inertia,  I,  which  for  reasons  to  be  dis- 
cussed in  Section  13,  was  considered  to  vary  as  d^A.  Letting  M 
represent  the  bending  moment  at  the  centroid  of  a section  due  to 

* Since  the  tests  described  herein  were  made  and  reported,  a 
very  general  theory  for  analyzing  frames  of  this  type  has  been 
published  by  G.  A.  Maney,  in  Trans.  A. S. C.E. , Vol.LXXXIII ,p664 , ^ 
1919-20 


5 


Fig.  1 


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for  the  horizontal  reaction  of  a two  hinged  arch  was  applied. 


H = 


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Prom  the  conditions  for  static  equilibrium  of  the  frame  the  value 
of  the  moment  at  midspan  was  found  to  be 

______  -(i^) 

Substituting  values  of  H from  equation  (1)  in  equation  (2)  values 


Me  = M --  Hh 
^ 6 


of  Mq  ?/ere  obtained,  and  for  convenience  have  been  expressed  in 
terms  of  the  maximum  moment,  , for  a simple  beam  loaded  as  the 
top  member  was  loaded.  The  reason  for  using  the  latter  quantity 
is  that  it  simplifies  the  application  of  values  of  M to  designing. 

To  illustrate  the  relations  expressed  by  equation  2,  let  it 
be  assumed  that  the  value  of  Me  is  known  for  the  rectangular  frame 
of  Pig.  2(a).  Since  the  horizontal  member  is  a partially  restrain- 
ed straight  beam,  the  sum  of  the  negative  and  positive  moments  is 

equal  to  , and  the  trapezoid  ABGD  represents  the  moment  diagram 
6 

for  a simple  beam.  Hence  laying  off  M„  fixes  the  value  of  the 
negative  moments  at  the  corners  and  the  entire  moment  diagram 
for  the  frame  is  easily  drav/n.  It  is  further  evident  that  if  R is 


Por  derivation  of  this  formula,  see  Johnson,  Bryan  and  Turneaure, 
"Modern  Pramed  Structures",  Part  II,  pages  138  to  158.  In  this 
derivation  the  effect  of  deformations  due  to  internal  shearing  and 
direct  stresses  are  neglected.  \^l/hile  these  effects  are  usually 
negligible,  they  may  be  included  in  equation  1 by  measuring  ^ 
to  a point  other  than  the  centroid  of  each  section.  Such  points 
may  be  determined  by  use  of  the  theory  of  the  ellipse  of  elasticity 


8 


the  resultant  of  the  horizontal  and  vertical  reactions  at  the 
base,  the  moment  at  any  point  left  of  ^ is  This 

relation  is  particularly  useful  in  treating  a frame  having  brackets 
in  which  the  axes  of  the  horizontal  and  vertical  members  do  not 
meet  at  right  angles,  as  in  Fig,  2fb).  Here  the  top  member  is  not 
straight  and  the  sum  of  positive  and  negative  moments  is  not  equal 


to, 


However,  it  is  possible  to  lay  off  the  knom  value  of  _M, 


and  the  simple  beam  moment  diagram  ABCH.  This  determines  the 
moment  diagram  for  the  horizontal  and  vertical  portions  of  the  . 
frame  and  locates  the  point  of  inflection  0,  The  resultant  re- 
action R must  pass  through  0 and  since  the  moment  at  any  section 

j 

to  the  left  of  B varies  as  the  distance  from  R,  the  moment  dia- 
gram for  the  corner  of  the  frame  can  be  drawn  in  ijroportion  to  the 
moment  in  the  vertical  portion  of  the  frame. 

Fig,  3 shows  the  results  of  analyses  made  by  use  of  equation 
(1)  for  the  purpose  of  designing  test  specimens.  Values  of  the 
bending  moment  at  midspan  are  plotted  as  ordinates  against  hori- 
zontal lengths  of  brackets  as  abscissas.  The  points  representing 
the  calculated  moments  are  seen  to  lie  nearly  on  a straight  line. 
The  variation  in  moment  at  midspan  is  due  almost  entirely  to  the 
variation  in  stiffness  produced  by  the  different  lengths  of  bracket. 
It  is  rather  surprising,  hov;ever,  that  the  moment  should  vary  so, 
nearly  as  a linear  function  of  the  bracket  length,  and  hence  of 
the  clear  span,  for  these  particular  frames.  From  the  difficulty 
of  analyzing  such  a frame  a straight  line  relation  would  not  be 
expected  to  obtain,  and  it  is  evident  that  the  line  does  not  apply 
exactly  at  the  * two  extremities  of  the  diagram  where  the  bracket  is 


9 


Fi^.  3.  Relation  between  Calculated  Moment  and  Size  of  Bracket 


Tip 


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10 


either  very  small  or  very  large. 

To  determine  whether  similar  curves  may  he  drawn  for  frames 
of  other  proportions  further  calculations  have  been  made.  Fig.  4 
indicates  the  relative  effectiveness  of  brackets  when  used  in 
frames  having  different  ratios  of  height  to  span.  It  is  seen 
that  the  use  of  a given  bracket  has  the  greatest  effect  in  chang- 
ing the  value  of  the  moment  at  midspan  when  the  ratio  of  height 
to  span  of  the  frame  is  small.  This  is  apparently  due  to  the 
fact  that  as  the  ratio  of  height  to  span  decreases  the  bracket 
occupies  a larger  portion  of  the  region  of  high  negative  moment. 
The  principle  involved  here  is  sufficiently  important  to  warrant 
further  elaboration.  For  illustration,  if  in  any  frame  the  moment 
of  inertia  for  a short  portion  of  length  be  changed,  there  will 
be  a change  in  the  bending  moment  at  any  other  point  in  the  frame. 
This  change  is  approximately  proportional  to  the  original  bending, 
moment  at  the  section  where  the  change  in  moment  of  inertia  is 
made.  It  will  be  seen,  therefore,  that  an  increase  in  the  moment 
of  inertia  such  as  that  produced  by  a bracket  or  haunch  will  be 
most  effective  if  made  where  the  original  bending  moment  is  larg- 
est. Referring  again  to  Fig.  4,  when  the  ratio  of  height  to 
span  is  1.0  the  negative  moment  at  the  corner  of  a frame  without 
brackets  is  .40  and  a 24  in.  bracket  reduces  the  moment  at 
midspan  only  20  per  cent.  On  the  other  hand,  when  the  ratio  of 
height  to  span  is  0.3,  the  negative  moment  at  the  corner  of  the 
frame  without  brackets  is  .56  ^ and  the  24  in.  bracket  reduces 
the  moment  at  midspan  by  51  per  cent.  Hence  for  brackets  to  be 
most  effective  both  in  changing  moment  distribution  and  in  reduc- 


11 


jE^oc^:  o:^  i^xobox^rTOUS  o:^  LXOTJe 


ing  stresses  they  must  he  used  at  points  of  high  bending  moments. 

Another  series  of  calculations  was  made  to  investigate  the 
effect  of  slenderness  of  members  on  the  effectiveness  of  haunches. 
With  a constant  ratio  of  height  to  span  and  a constant  ratio  of 
bracket  length  to  depth  of  member,  5’ig*  5 shows  that  the  effect- 
iveness of  a bracket  in  reducing  the  moment  at  midspan  is  less 
for  the  smaller  depths  of  member.  This  should  be  true  because 
the  frame  is  stiffened  along  a smaller  portion  of  its  length,  and 
because  the  change  in  shape  of  the  axis  of  the  frame  at  the 
corners  is  less  for  the  slenderer  member.  The  information  of 
Fig.  4 and  Fig.  5 has  been  replotted,  using  values  of  the  moment 
at  midspan,  Mq,  as  ordinates  and  the  ratio  of  the  clear  span  to 
the  total  span,  s/^  , as  abscissas,  producing  the  curves  shown 
in  Fig.  6,  Each  curve  represents  a certain  value  of  h/£  , the 
ratio  of  height  to  span.  The  variation  in  clear  span  indicated 
in  this  diagram  is  obtained  by  varying  both  the  size  of  the 
brackets  and  the  slenderness  of  the  members  of  the  frame. 

This  diagram  indicates  that  the  moment  at  midspan  varies 
very  nearly  as  a linear  function  of  the  clear  span,  or  distance 
between  bracket  edges,  for  frames  of  the  proportions  shown.  That 
is,  just  as  in  the  case  of  the  curve  of  Fig.  3,  here  a series  of 
straight  lines  seem  to  fit  the  several  groups  of  calculated  points 
fairly  well,  the  divergence  from  the  linear  relation  being  shown 
at  the  extremities  of  the  lines  by  dotted  curves.  The  straight 
portions  of  the  curves  represent  the  range  of  values  ordinarily 
encountered  in  the  use  of  brackets;  wi thbra'- ]^ets  so  small  that  the 
ratio  s/^  becomes  0.9  or  more  further  tests  are  needed  to  deter- 


0.6 


15 


Id 


O 


Q> 


JO  S'LUJSJ.  U!  uod^pij^ ^.O^l^JU^LUO!^ 


06  0.8  lO  /.2  !4 

Depth  of  mem ber,  d,  m Feet 

Fig. 5,  Relation  between  Slenderness  of  Frame  and  Calculated  Reduction  in  Moment  Due  to  Bracket. 


14 


E’ig.  6,  Effect  of  Length  of  Clear  Span  upon  Moment  at  Midspan. 


15 


mine  the  exact  effect  upon  the  moment  distribution. 

It  is  significant  that  for  the  ratios  of  depth  of  span  of 
members  to  be  found  in  practice,  the  moment  is  not  greatly 
affected  by  a variation  in  slenderness  of  members  as  long  as  the 
clear  span  is  not  changed.  This  indicates  that  the  clear  span 
is  the  variable  of  major  importance.  The  effect  of  a variation 


in  slenderness  is  still  less  v:ith  higher  values  of  n than  that 
shown  with  values  of  n eg^ual  to  0.25  and  3/7. 

It  is  thought  that  an  ec[uation  representing  the  curves  of 
Fig.  6 may  be  found  useful.  Such  an  equation  must  naturally 
reduce  to  the  ordinary  equation  for  rectangular  frames  when  no 
brackets  are  used.  If  n represents  the  ratio  h/g  for  such  a 
frame  without  brackets,  the  moment  at  midspan  due  to  a total 
load  P applied  in  equal  parts  at  the  1/3  points  of  the  span  is 
expressed  by  the  equation* 


Me  = 


2n  + 1 
2n  + 3 


(3) 


ITow  ivith  regard  to  frames  with  brackets**,  it  is  found  that  letting 


* For  demonstration  leading  to  this  equation  see  Bui. 108, 

Eng.  Expt.  Sta.,  Univ.  of  111.,  1918,  p.56» 

**A  method  of  analyzing  frames  similar  to  these  is  given  by 
E.  Bjornstad,  in  "Die  Berechnung  von  Steifrahmen  nebst  anderen 
statisch  unbestimmten  Systeraen.”  Berlin, 1909.  One  equation  is  used 
for  frames  both  with  and  without  brackets  by  proper  choice  of  the 
terms  corresponding  to  n.  That  is,  the  members  which  contain 
brackets  are  considered  replaced  by  ’’equivalent"  members  of  con- 
stant cross-section  throughout.  The  section  of  the  equivalent 
member  from  which  n is  calculated  obviously  varies  with  the  size  of 
bracket.  This  treatment  of  the  subject  from  a purely  theoretical 
vie?;point  neglects  the  change  in  shape  of  the  axis  of  a frame  con- 
taining brackets;  further  the  assumption  is  made  that  for  a haunch 
which  varies  uniformly  in  depth  the  moment  of  inertia  also  varies 
as  a linear  function  between  the  two  extremities  of  the  haunch. 

The  results  obtained  are  stated  to  be  approximate. 


16 


m = s/^  , the  ratio  of  clear  span  to  total  span,  the  straight 
portion  of  the  group  of  curves  of  Fig.  6 may  all  be  expressed 
by  the  equation 


- 

6 


2n  + 1 0.65  - 0.7  m 

2n  + S 


(4) 


nii"  +”  0.8 

This  is  an  empirical  equation  fitted  to  the  results  of  semi- 
graphical  analyses  and  hence  has  a theoretical  basis.  Its  use, 
however,  should  be  limited  to  values  of  m between  0.5  and  0.93 
and  to  values  of  n between  0.3  and  2.0.  It  may  be  noted  that  for 
any  particular  value  of  n the  effect  of  all  sizes  of  brackets  is 
determined  by  calculating  t?/o  values  of  from  equation{4][ , since 
these  are  sufficient  to  fix  the  position  of  a straight  line  simi- 
lar to  those  of  Fig.  6. 

The  foregoing  analysis  has  been  based  upon  the  use  of  equal 
moments  of  inertia  of  columns  and  girder  except  at  sections 
occupied  by  brackets.  In  practice  it  is  quite  likely  that  the 
columns  and  girders  of  a bent  may  have  considerably  different 
cross-sections,  with  a resulting  variation  in  the  moments  of 


inertia.  For  columns  without  brackets  equation  (3)  still  applies 

lih 

ZT  ■ 


if  the  term  n is  considered  equal  to  'J-  , fig  = moment  of 


inertia  of  girder  section,  1^  = moment  of  inertia  of  column  sec- 
tion) . It  is  therefore  immaterial  whether  a variation  in  n is 
caused  by  a variation  in  the  ratio  of  height  to  span  or  of  moments 
of  inertia  or  of  both.  However,  when  brackets  are  used,  such  a 
general  relation  apparently  does  not  obtain. 

A number  of  calculations  have  been  made,  using  a constant 


value  of  h , and  of  moment  of  inertia,  lo-,  of  girder,  but  varying 

-e  ^ 


*'.C- 


■’■Jm 


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■ •d: 


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■'4?^ 

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TC^  ':  r57;5(J)0<::  |f 


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i T’-'  i a?  ; ..  , M • 
' • ' .'• ' 

'•:t.‘  ’.  ' .2 L . 

or.-;-  ■ ,/r.  0 ci.  : ;!i'c 


i.Tj  ,:0A'  ,*y  -v7  4 

A . -r*  ' ' 

LU.  rt#,  , . f-^/rrt  , ,.^  ■ (*)  'X^. 


P'S 


‘j  t- 


!i  ■ i-t'-'l’-tr:,  r iT 

■ i -: ",  oX.-io  lo  *r  • 


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17 


the  moment  of  inertia  of  the  portion  of  the  columns  below  the 
lower  edge  of  the  bracket*  The  values  of  thus  determined  are 

plotted  against  values  of  the  quantity  Zg  in  Pig*  7.  It  will 

be  noted  that  with  values  of  ^ equal  to  3/7  and  1,  respectively, 
widely  different  values  of  Mq  are  found  with  the  same  value  of 
Furthermore,  these  values  do  not  compare  at  all  closely 


is 


With  the  moments  obtained  by  assuming  ig  . h equal  to  n in 

^c  T 

Equation  (4).  It  seems  impracticable,  therefore,  to  attempt  to 
use  Equation  (4)  to  investigate  frsmies  in  which  the  moments  of 
inertia  of  girder  and  columns  vary.  Pig.  7 indicates  the  general 
range  of  values  of  moments  to  be  obtained  in  most  cases  of  this 
sort;  frames  of  other  proportions  may  be  analyzed  by  the  applica- 
tion of  Equation  (1). 

It  has  been  shown  that  the  moment  at  midspan  decreases  with 
the  increase  in  stiffness  at  the  corners  of  the  frame  produced 
by  the  use  of  brackets.  If  the  corners  of  a rectangular  frame 
could  be  stiffnned  x7ithout  the  use  of  brackets,  a decrease  in 
moment  at  midspan  would  cause  an  equal  increase  in  moment  at  the 
comers,  since  in  this  case  the  numerical  sum  of  the  positive  and 
negative  moments  is  equal  to  the  maximimi  moment  for  a simple 
beam  carrying  the  same  vertical  loading.  However,  where  brackets 
are  used  the  axes  of  the  members  do  not  intersect  at  right  angles, 
but  approach  more  nearly  the  pressure  line  of  the  resultant  force 
acting  on  the  hinge.  The  resulting  decrease  in  moment  at  the 
corner  due  to  this  variation  in  shape  of  specimens  with  brackets 
approximately  offsets  the  increase  in  moment  at  the  comer  due  to 
the  variation  in  stiffness  of  the  different  parts  of  the  frame. 


p 

*'  . > 


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' . ''  ' 

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f-  ■' 

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18 


Pig. 7*  Calculated  Moments  in  Frames  of  Varying  Moments  of  Inertia 


19 


Hence,  the  negative  moments  in  the  different  types  of  frame  are 
nearly  equal  in  magnitude  for  a given  load,  as  will  he  noted 
later  from  the  results  of  tests, 

of  Brackets  with  Different  Loadings.-  Equation  (4) 
has  been  developed  for  the  special  case  of  third-point  loading 
which  was  used  in  the  test  frames.  This  form  of  loading  is  fre- 
quently used  in  tests  because  it  is  easy  to  apply  and  produces 
a moment  diagram  somewhat  similar  to  that  due  to  a uniformly  dis- 
tributed load,  A study  of  the  effect  of  other  loadings  shows  that 
equation  (4)  may  be  adapted  to  a form  which  gives  the  effect  of 
brackets  for  such  cases. 

Influence  lines  for  the  horizontal  reactions  of  a two-hinged 
frame  under  vertical  loads  are  shown  in  Pig*  8.*  That  is, 
the  ordinates  of  any  points  on  the  influence  line  represent 
relative  values  of  the  horizontal  reactions  due  to  equal  vertical 
loads  at  the  corresponding  points  of  the  span.  These  relative 
values  are  independent  of  the  ratio  of  height  to  span  of  the 
frame.  Prom  these  curves  it  is  found  that  certain  common  types 
of  loading  (with  equal  total  loads)  produce  the  following  relative 
values  of  the  horizontal  reaction ,H, considering  the  reaction 

* The  construction  of  these  influence  lines  is  based  upon  the 
follOY/ing  theory:  With  a frame  of  the  type  used  in  this  investiga- 
tion suppose  outward  thrusts  to  be  applied  at  the  hinges,  causing 
the  top  member  to  deflect  down\7ard,  How  from  Maxi-Bell’s  theorem 
of  reciprocal  displacements  it  is  known  that  the  elastic  curve  of 
the  top  member  of  this  frame  is  an  influence  line  for  the  horizon- 
tal reactions  of  a similar  frame  lo  ded  with  vertical  loads  on  the 
top  member  and  having  the  hinges  at  the  base  held  stationary. 

Pig. 8 therefore  was  obtained  by  computing  the  shape  of  the 
elastic  curves  of  t\No  extreme  forms  of  top  member;  one  having  no 
bracket  and  one  having  large  45°  brackets  extending  to  the  quarter 
points  of  the  span.  It  is  to  be  noted  that  these  curves  give  only- 
relative  (not  absolute)  values  of  the  horizontal  reaction,  and  that 
the  curves  can  not  be  used  to  compare  values  of  the  reactions  for 
the  two  types  of  frame.  However,  relative  values  are  sufficient  to 
compare  loads  at  different  points  on  the  same  frame,  which  is  the 

purpose  of  this  diagram. 


r . 


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*•  .'tgX-t  ' •■ 

• • • ‘ A 

' ... .. . V ' ^ ■■  > - 

".kV  / 

,!  e>  .‘i’dv' 

0’^ 

^rot  f 

, f'  ’T  -••  . > 

"’■:.  I',  ■'  n'.rov 

V*  • 

. ? ri  A>  «c  '1  ■ 

'th  L ]>•  ' ', 

■ TO  ,i  V>.  . 

';■<  .iw'ij  ij.'.-,:.:,  ■ • / ■ ; 

'.  li  4 , 

■ _ ■_.  '•■*  .; ' 

:-:xT'.:r‘v3  *)«  X f 

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rv  -V 

• ^ 4 ' . 

d..  ' fri.'.:/  : ■ • id 

.0.CO.-?  ;,.|  ;jufv  .t  ; 

t>;;’  I’d' 

•I  ' I i-  ' ■ 

..d  ^ 

BfUi 

^ - j:  ■ ' > 

■'•'  'ifa  ■ - i i, 

' w w 

• ;-ha  ■ -.r ; ■ ‘ ’I'M 00  -^-.f 

jM/d  1 

Cl-,  ■>'*;  c-': 

rft 

,o"v  i ' 

■ '“T.^  ; ? •*  d 

..'’rt  ■ c.'d 

’■'  ft  -•  7 

,.0  : f ?4A~'  Y 

i: 

'd-'x 


* * 


• /.to'.  '■]••.  .r  t .•  tori) 

^X'^;x,  j I'ix  V’,;  j; ... ,^;ot  '£ios<r  . ■;  d-.  -n*- ^ 

■ ^ _ jjjj  -;;  gr^  .'iBij 

Afc  •'  ■ '.•:'v  : r ZV 


. ' . ,i  U/.'  ^ .1  ^ ■■-  Y»  Cff 

■‘•-•ti-^oj:.  ’ • ■' 

•\  ' ' , .T^  : ; .. 


. \ ’ V ; ••  7':“' 

^ iuliLlKKe  ^0',.mrrC^. 


20 


Fig.  8.  Influence  Lines  for  Horizontal  Reactions  of  Frames. 


El 


due  to  third-point  loading  as  unity. 


Third  Point  Uniform 
Loads  Load 

No  bracket,  H = 1,00  H = .750 

Large  Bracket,  H = 1,00  H = ,7E0 


Concentrated 
Center  Load 

H = 1.125 
H = 1.167 


It  is  evident  that  the  relative  values  of  H for  a frame 
Tvith  no  brackets  will  not  be  in  error  more  than  4 per  cent  if 
used  for  frames  with  brackets. 

Prom  equation  (2)  and  (4),  for  the  third  point  loading. 


H = 


Pi 

6h 


En  + 5 


0,65  - 0,7m 

n^  + 0,8 j 


(5) 


Using  the  above  ratios  of  H for  the  fr^me  without  brackets 
and  solving  for  M , gives 

uniform  load  W,  Mq  = 


fEn 

+ 1 

0.65 

- 0.7m  / 

8 

l2n 

+ 3 

n^  + 

1 

1 

cc 

. 

o 

Me  = 

u 

En  +1.5  _ 

0.49  -0.53m 

4 

En  + 3 

n^  + 0,8 

— (7) 


For  any  other  type  of  loading  the  relative  values  of  H may  be 
found  from  Fig.  8, after  which  the  moment  is  readily  computed. 

5.  Effect  of  Brackets  in  Various  Types  of  Frames.-  While 
the  results  of  the  foregoing  analyses  should  be  useful  in 
the  design  of  two  hinged  frames,  the  question  immediately 
arises  as  to  what  quantitative  application  may  be  made  to 
other  kinds  of  frames.  For  example,  brackets  may  well  be  used 
in  the  closed  rectangular  frame,  in  the  two-legged  frame  with 
column  bases  fixed, in  building  or  viaduct  frames  and  similar 
stride  tu  res.  In 


E2 

the  absence  of  test  data,  theoretical  analyses  have  been  made  for 
two  types  of  frame  and  it  appears  that  an  application  of  ecLuation 
(4)  may  be  made  to  these  cases. 

The  frames  analyzed  are  the  closed  rectangular,  or  quad- 
rangular, frame  in  which  the  columns  are  assumed  to  be  rigidly 
attached  at  the  bases  to  a horizontal  member  of  equal  section, 
and  the  two-legged  frame  with  the  columns  fixed  at  the  bases. 

Both  of  these  frames  have  points  of  contraflexure  in  the  columns 
at  a distance  ho  equal  to  2/3  h or  more  from  the  tops  of  the 
frames.  In  this  case  the  portion  of  the  frame  above  these  points 
of  contraflexure  may  be  considered  as  a two-hinged  frame  in  which 
the  bases  have  been  alio  /ed  to  move  apart  a small  amount. 

Such  an  outward  movement  tends  to  increase  the  moment  at  midspan 
in  a two -hinged  frame. 

The  analysis  of  these  frames  has  been  made  by  use  of  the 
general  theory  of  indeterminate  structures,*  The  span  and  depth 
of  members  used  were  the  same  as  those  used  in  the  test  specimens, 
while  the  height  and  clear  span  of  the  frames  were  varied. 

Fig,  9 shows  values  of  the  calculated  moment  at  midspan  for 
varying  proportions  of  these  frames,  and  compares  them  with  values 
for  the  hvo-hinged  frame.  It  is  seen  thi t for  frames  having  the 
same  distance,  hQ,  from  the  top  to  the  point  of  contraflexure, 
the  moments  M are  very  nearly  the  same.  The  values  for  the 
fixed  base  and  quadrangular  frames  are  slightly  higher  than  for 
the  two-hinged  frame  because  the  columns  at  the  point  of  contra- 


* See  Johnson,  Bryan  and  Turneaure,  Modern  Framed  Structures, 
Part  II,  pp,386 


1.. 


tit* 


^w:  ■ r 


I f 


-‘AC  ^3:*:v;.f. 
V*  ’ I 0 j j • T 


- -. 


■ y 


• - ; , '-'M 


•vt.lv  n ^ 


'v.i  "vi' '"i;.'. . 

■Jj  'V  : I \ ' \r,'  'uO'>  7 i ' r 

■ ^ ■ ' . ■ 9 ;>  7 

i:*'  • ,C;  ':i  ’ ' 

'/  1 7 

•*  ■' ' ' A 


i#  ^ 


’ T 


' r . , X 

.■,^.-d  ‘.\:  .-!•  i 

■ u ‘UK*dJ  t.  - ^ ■"' 

r,'  :.*•  ,.  _ ‘'.*i*';:.iB..'  j f. . 


' X j - 1 ■ T '.'  1 • ' ; ;J 

• ■■  •!  ii  ’ . 

fji''-*.'",..  j 

■.•  ■ -•:  f).,  ■ • 

' - 

'■■•■  .•'  ; i' 

,?(■.'  tRlw  ---  ■ *-;  . 

MV 

"0.:  'w 

X:  -ur 

V:  • • . . r . -•• 

i 

; 1 

'■  ■•■  • r'/,^ 

■ ■■  rr.^.’T-'  ; 

" ■'■  ' T ==  , 

• : ;> : i- .^qo  n 

.'■i  ■ 


' **  C!  ■ I 


; -O  O . .' 

'1 ';0^  h'v  ^ :ti'  . 


^ • * 

V I ;*  ' 


rt^  y - n: 


V I T 


' ■*-'  .' 

V I'j  r’ji-  r .i/'; . i;,,-.  . 

> ' - ^ "i  :.i5;.v  •.;'!:  r,i: 

, y j .:-  ^ • : : ' f-' 

■ c:  . '.;  •v-’. . ■ • ' ■ ^ ('rj  L : 

0 . .f  ?. 


;i:;  ; o. 


iril 

fiT. 


?.  ■•  ' A 


V'v  ■,  A 


r tJ*' 


1 , 10':: 


•■■’  :r.  . 'X  ri 


'li"'  Af;.in.rf£^4Mp  'SritS^l 

'1 


i'-''  •'■ 

4 • I ^ • 


'j-  • -11  ■t--.;C0!. 


. "r 


ri  v:r' 

Ax: . 


oyT 


V • 


^ Tt.  , 


- ir  . \,  1'.’-! 


Moment  of  M/depon  /n  terms 


Fig. 9. 


Moments  at  Midspan  of  Frames  of  Different  Types  and  Proportions 


Types  of  Frames 
and  corresponding  symbo/s 
Depth  of  members  /-o" 

, , Jpan  /d'O" 

^ l \*~  I F 

T I 

_L 

X— X— X 0-0-0 

, I 

d/nged  f/xed  Ooadrangu/or\ 

I 

.J  d 5 .6  .7  T'  3 LG  // 

Va/ims  . * 


ss 


I •• 


)'  (> 


r -..♦s 

..  - 


(■ . * 


i 


it 


I 


■ f 

t'f  ti-.  r. 


'!« . 


I « 
8'  • 


l-v-.!: 


! i^i-’ 


I 

r 


1 X]  v^'vi  . ■ -s.^ - ^ ik'  Vxx ■’: 


I',  j 

I r^-'  VH  ' h~  ^ J }^  JiH 

( rr~n  ^ 


i 

j i~ 


,,  , •: 


■c.  J1.X: ?■■'•■■ ..  ■ - ‘’^  ■■  ,f  ■»' 


r'  '■  ?/j-, 

...>x"f 


■ ’■-  -r^i 


» 


<f' 


B«g^  *):o  5;/^  irf.cfsfciM  ^ja  a^iiemolT.  .^.^ilT 

‘ ' - '‘-  ■ '•■.■  . \ .5.V.-  #:  .,  ''^  ^ '*•!  ■• 


'■  I- 


flexure  have  deflected  outv/ard  slightly.  However,  if  the 
value  of  ho  could  be  determined  it  aj^ears  that  these  frames 
could  all  be  treated  as  two  hinged  frames  by  use  of  equation  (4  ) 
without  appreciable  error,  For  this  purpose  Pig,  10,  giving 
values  of  h^,  for  frames  of  various  proportions,  may  be  used. 

The  data  for  this  diagram  were  computed  along  with  those  for 
Pig.  9. 

The  procedure  in  determining  the  effect  of  certain  brackets 
in  a quadrangular  or  a "fixed-base”  frame  may  be  summed  up  as 
follows: - 

Knowing  the  proportions  of  the  frame  the  position  of  the 
points  of  contraflexure  may  be  determined  from  the  curves  of 
Pig,  10,  Placing  the  value  of  h^^  equal  to  n in  equation  f4) 
gives  the  value  of  Mq,  the  moment  at  midspan  of  the  top  member. 
The  moments  at  all  other  parts  of  the  frame  may  be  found  from 
the  eqtiations  of  statics,  ^'Thile  the  value  of  ^ may  be  slightly 
inexact,  the  error  is  small. 

The  curves  giving  the  position  of  the  points  of  contra- 
flexure are  applicable  for  any  loading  on  the  top  member  that  is 
synmetrical  about  the  center  line  of  the  frame.  Hence, if  uniform 
or  concentrated  loads  are  used  the  procedure  is  the  same  except 
that  equations  (6)  and  (7)  must  be  nsed  to  determine  the  value 
of  Me* 

One  or  two  other  trypes  of  rigidly  connected  structures 
may  well  be  mentioned  here,  Por  example,  the  continuous  beam  of 
three  spans,  with  loads  on  the  middle  span  only,  may  be  treated 


25 


<D 

u 

M 


jH 

o5 


4^ 


o 

o 


tH 

O 

-P 


o 

P4 


O 

4-» 

(1) 

0 


o 


ft 

o 

EH 


0 

o 

h 

O 


Q) 

O 


0} 


-P 

CQ 


•H 

P 


O 

P 


bD 

•H 

ft 


26 


as  a two-legged  frame  with  the  coltimns  swimg  up  into  the  hori- 
zontal position.  liThile  certain  features  of  the  rectangular 
frame  such  as  column  action,  curved  beam  action  and  direct 
compression  due  to  horizontal  thrust  are  absent  in  the  continu- 
ous beam,  brackets  used  at  the  supports  will  occupy  the  same 
region  of  bending  moment  in  the  two  tjrpes  of  structure.  Hence 
equation  (4)  will  give  substantially  correct  results  when  used 
with  this  special  case  of  a continuous  beam. 

Another  very  common  form  of  continuous  beam  is  one  having  a 
large  number  of  spans  of  identical  dimensions  and  with  all  spans 
subjected  to  equal  loads.  To  illustrate  the  effect  of  brackets 
used  at  the  supports  in  such  a beam  two  numerical  cases  have  been 
analyzed  by  the  method  explained  at  the  beginning  of  this  section. 
In  one  case  brackets  with  a 45°  or  1 to  1 slope  were  used  at 
each  side  of  the  supports;  in  the  other  case  brackets  having 
a 2 to  1 slope  (making  an  angle  of  26^32'  v/i/th  the  horizontal) 
were  used.  The  ratio  of  depth  to  span  was  one  tenth,  though 
this  quantity  is  shown  in  Pig.  6 to  be  relatively  unimportant. 

Pig,  11  shows  values  of  the  moment  at  midspan  due  to  one- third 
point  loading,  for  various  values  of  the  ratio  of  clear  span  to 
total  span*  Denoting  this  ratio  by  ra  it  is  found  that  approximate 
formulas  for  the  moment  at  midspan  are; 

Por  brackets  with  1 to  1 slope. 


Por  brackets  with  2 to  1 slope. 


The  moment  over  the  supports  in  either  case  is 


27 


03 

■P 

M 

o 

o3 

u 

W 

-P 

•H 

03 


® 

PQ 

OQ 

pi 

O 

pi 

Pi 

•H 

+3 

Pi 

O 

o 

O 

PJ 

o3 

ft 

03 

fCJ 


■P 

etJ 

03 

p» 

Pi 

© 

S 

p 


to 

•H 

ft 


i 

i 


28 


Ms  Me --(10) 

Equations  (8),  (9)  ^.nd  (10)  may  "be  applied  to  the  case  of  a 
iiniformiy  distributed  load  W on  each  span  by  replacing  the  quantity 
^ * These  equations  should  not  be  used  for  values  of  m 

smaller  than  thos  e shown  in  Pig.  11. 

The  diagram  shows  clearly  that  with  very  large  brackets, 
corresponding  to  small  values  of  ra,  the  bracket  carries  nearly  all 
of  the  load  directly  to  the  support,  acting  as  a very  stiff  canti- 
lever. The  positive  moment  at  midspan  accordingly  becomes  very 
small . 


I 


29 


III.  TEST  SPECIMENS  AND  APPARATUS 
6.  Description  of  Test  Specimens,-  Test  specimens  of 
five  types  v/ere  used;  two  specimens  each  of  Types  A,  B,  C,  and 
D,  and  one  of  Type  E,  All  were  two-legged  frames  with  the 
colnmns  hinged  at  the  bases.  The  details  of  these  specimens 
are  indicated  in  Pig.  12  to  20.  The  cross-sections  of  types 
A,  B,  C,  and  D were  of  T- shape,  having  the  following  nominal 
dimensions;  depth,  12  in.;  thickness  of  flange,  3 in.;  v/idth 
of  web,  8 in;  width  of  flange,  30  in.  The  flanges  were  intended 
to  produce  the  effect  of  the  shell  of  a ship  adjacent  to  the 
frame.  The  cross-section  of  type  E was  rectangular,  being 
nominally  12  in.  deep  and  8 in.  wide.  This  type  furnished  a 
comparison  with  type  A to  show  the  effect  of  the  difference  in 
section. 

All  of  the  test  specimens  were  made  15  ft.  long  and  7 ft.  high, 
overall.  The  span  from  center  to  center  of  hinges  was  14  ft. 
and  the  height  from  center  of  hinges  to  mid-depth  of  girder  was 
6 ft.  Actual  dimensions  differed  slightly  from  these  nominal  fig- 
ures. 

Specimens  of  Types  A and  E were  made  with  square  corners  at 
the  intersection  of  horizontal  and  vertical  members  with  the 
exception  of  a 2-in.  fillet  at  the  interior  corners.  These 
fillets  were  used  to  modify  the  extremely  high  stresses  which 
occur  at  a sharp  corner,  but  were  not  regarded  as  capable  of 
exerting  any  appreciable  effect  as  brackets. 

Types  B,  C and  D were  made  v;ith  brackets  similar  to  those 
actually  used  in  concrete  ship  construction.  Type  B had  45° 


1 .1'  n ■ I G 

. ••  1 > 


r 

I 


■■^r-  ■ i.’vvi’r 

: ' . ' ' •,«-•■  .k 


30 


r 

! 

1 *? 

;--a 

lU 

0 



<D  i 1^ 

7' 

, \ 

1 1 

1 1 

F=i r-i!  1 ^ 

7 -K 


3 

I / 


Fig,  12.  Details  of  Test  Specimen  13A1 


^ou'i'h  Z^'cufh 


31 


V d 


i ? 


<o 

4 


c .4*  J 


'i 


rr. 

A 

•'  / 


V 


•m. 


' .-  t'Z 
„ rv  c?r 


J, 


Fig.  13.  Details  of  Test  Specimen  13A2 


Fig.  14.  Details  of  Test  Specimen  13B1 


U TJ 


S>ouih  5outh 


36 


Fig.  18,  Details  of  Test  Specimen  13D1 


5oui'h  S>ouih  A/orth 


37 


Pig,  19.  Details  of  Test  Specimen  13D2 


S>oufh  5c>t//A  Horj-r< 


38 


Fig.  20.  Details  of  Test  Specimen  13E1 


59  5' 

■brackets,  12  in*  in  horizontal  length;  the  exterior  corners 
were  given  a 45®  chamfer  eQ.Tial  in  size  to  the  bracket,  making 
the  depth  of  cross-section  noimal  to  the  face  of  the  bracket 
about  17  in.  Type  C was  similar  to  Type  B,  but  had  a bracket 
24  in.  in  horizontal  length,  making  the  depth  of  cross-section 
at  the  corner  approximately  25  in.  Type  I)  was  modified  from 
T3rpe  C by  filling  in  the  angles  bet;veen  the  bracket  and  the 
main  members  i^lth  two  supplementary  haunches,  so  that  the  inside 
line  of  the  frame  approached  the  outline  of  a curved  soffit. 

The  hinge  detail  at  the  bs^e  of  each  column  ?;as  provided  by 
casting  in  place  a steel  shoe  formed  by  5/4  in.  bearing  plates 
with  a 5 in.  pin-hole  at  each  side  of  the  column  connected  by 
a 5 in.  pipe  sleeve  28  1/2  in.  long.  A 2 15/16  in.  steel  pin 
passing  through  the  pinlioles  and  pipe  sleeves  engaged  similar 
plates  on  the  test  base  and  formed  a simple  hinge. 

In  the  design  of  the  rei:of orcement  used  in  the  test  speci- 
mens the  working  stresses  assumed  were  16  000  lb.  per  sq..  in 
for  tension  in  the  steel,  and  1500  lb.  per  sq.  in.  for  compression 
in  the  concrete.  The  ratio  of  the  modulus  of  elasticity  cf  steel 
to  that  of  concrete  was  taken  as  8.  The  design  provided  for  a 
reversal  of  the  direction  of  loading,  so  that  all  sections  con- 
tained a large  percentage  of  compression  as  well  as  tension  steel. 
Details  of  the  reinforcement  used  in  the  specimens  are  shown  in 
Pig.  12  to  20. 

In  designing  the  members  the  approximate  bending  and  resist- 
ing moments  were  calculated  and  the  section  at  which  failure  by 


40 

compression  would  probalDly  occur  was  determined,  From  the 
resisting  moment  in  compression  at  this  point  the  \Yorking  load 
for  the  specimen  was  calculated  and  sufficient  tension  reinfoice- 
ment  was  provided  at  all  points  to  withstand  the  external 
bending  moment. a The  specimens  were  heavily  reinforced  with 
bent  bars  and  stirrups  against  diagonal  tension  failure. 

Table  1 shows  the  percentage  of  longitudinal  steel  used  in  all 
specimens,  based  6n  the  area  of  oross-section  exclusive  of 
flanges. 

TABLE  1. 

PERCENTAGE  OF  LONGITUDINAL  RE  ECFORCEIvIEl^T 


Location 

Stress 

Spe  c imen 

Numbers* 

of 

in 

13A1-: 

2 13)B1-E 

13C1-2 

13D1-2 

Section  Reinforcement 

13E1- 

(Dovnward  Loads) 

Center  of 

^(Tension 

4.67 

4.67 

4.67 

4.67 

Girder 

(Compression 

5.61 

7.48 

3.74 

3.74 

Corner  of 

_ (Tension 

4.  67 

3.79 

2.05 

2.05 

Frame** 

(Compression 

5.61 

4.41 

2.46 

2.46 

The  main  reinforcing  bars  were  all  1 in.  plain  round  bars, 
and  the  stirrups  ?;ere  either  1/2  or  5/8  in.  plain  round  bars. 

The  tee  flanges  were  reinforced  with  cross  rods  to  distribute 
the  stresses  and  to  prevent  transverse  curvature  of  the  flanges. 

* This  investigation  was  performed  as  Test  Series  13; 
hence  the  series  number  is  used  as  a part  of  all  specimen 
numbers. 

**Vertical  section  for  Types  A and  E;  section  normal  to 
face  of  bracket  in  Types  B,  C and  D. 


41 


7.  Materials  and  Making  of  Specimens. ~ Cemen t. - 
Lehigh  Portland  cement  was  used  in  the  making  of  all  the  test 
specimens.  It  passed  the  requirements  of  the  United  States 
Government  Specifications  for  Portland  Cement.* 

Aggregate."  The  sand  and  gravel  were  obtained  from  local 
deposits  at  South  Bethlehem,  Pennsylvania.  The  material  was 
siliceous,  clean  and  gritty.  It  was  carefully  separated  by 
screening  into  three  sizes;  (l)  fine  sand  consisting  of  grains 
smaller  than  l/8  in.  in  diameter,  (£)  coarse  sand  falling 
between  l/8  in.  andl/4  in.  in  diameter,  and  (3)  gravel  exceed- 
ing l/4  in.  but  less  than  l/2  in.  in  diameter.  The  separation 
of  the  sand  into  fine  and  coarse  grades  was  clone  for  the  pur- 
pose of  controlling  the  imifomity  of  the  concrete  made  from  it. 

Steel.-  The  reinforcing  bars  were  re-rolled  from  rejected 
shrapnel  steel  billets  of  hi^  yield  point.  The  physical 
properties  of  this  steel  are  shown  in  Table  2.  Each  value 
in  the  table  is  the  average  of  two  tests. 

TABLE  2. 

PHYSICAL  PROPERTIES  OP  STEEL  BilRS 


Liam. 

in. 

Yield  Point 
lb. per  sq.  in. 

Ultimate  Tensile 
Strength 
lb. per  sq.  in. 

Elongation 
in  8 in. 
per  cent 

Reduction 
in  area 
per  cent 

1/2 

63  800 

104  400 

16.6 

27.5 

5/8 

66  070 

108  440 

18.0 

42.2 

1 

55  800 

91  425 

20.5 

38.2 

* Circular  of  the  Bureau  of  Standards,  Ho.  33. 


M .uru  P U.7 


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42 


Concrete. - The  concrete  used  in  this  investigation  was 
mixed  a]c:proximate 3y  in  the  proportions  1:1:1.  The  actual 
proportions  hy  volume  were  as  follows:  cement,  1 part;  fine 
sand*  part;  coarse  sand,  .20  part;  gravel  .99  part.  In 
preparation  for  mixing  the  concrete,  each  kind  of  aggregate 
was  thoroughly  mixed  by  shoveling  and  determinations  of  the 
moisture  content  were  made  upon  samples  taken  at  random*  Enough 
water  was  added  when  mixing  the  concrete  to  make  the  total  water 
13  per  cent  of  the  ccmbined  weight  of  the  dry  materials,  thus 
producing  a rather  stiff  mixture,  considerably  drier  than  is 
generally  used  in  reinforced  concrete  construction  work. 

Auxiliary  Specimens.-  Six  6 x 12  in.  cylinders  were  made 
with  each  test  specimen  and  were  stored  under  the  same  conditions 
until  tested.  Three  cylinders  in  each  lot  were  tested  when  7 days 
old  and  the  remaining  three  were  tested  when  40  days  old,  the 
latter  being  the  approximate  age  of  the  frames  when  tested. 

The  average  compressive  strength  of  the  concrete  is  given  in 
Table  3.  The  initial  modulus  of  elasticity  of  the  concrete  was 
3/750  000  lb.  per  sq.  in.;  this  value  was  found  from  a large 
number  of  compression  tests  of  concrete  cylinders  having  identi- 
cal proportions  and  materials,  but  made  in  connection  with  other 
investigations  at  the  laboratory.  The  stress-strain  curves  for 
these  cylinder  tests  were  aloser  to  straight  lines  than  is 
usually  expected  with  concrete. 


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45 


TABLE  3. 

COILPHESSIVE  STRELGITH  OF  CONCRETE. 

Each  value  represents  the  average  of  three 
tests  on  6 X IE  in.  cylinders. 


Made  with 
Specimen 
No. 

Unit  Strength 
at  age  of  7 days 
lb. per  sq.  in. 

Unit  Strength 
at  age  of  40  days 
lb.  per  sq.  in 

13A1 

SEEO 

3110 

13AE 

S975 

3765 

13B1 

S880 

4S60 

13BE 

E535 

3665 

13C1 

S775 

4505 

13CE 

E900 

4040 

131)1 

E870 

44E0 

13  BE 

3075 

5860 

13E1 

S815 

4995 

Average 

S785 

4E90 

Making  of  Specimens.-  One  wooden  form  was  used  for  all 
test  specimens,  and  the  inside  corners  of  the  form  were  designed 


to  provide  for'  the  variation  in  shape  of  the  brackets.  The 
inner  surfaces  of  thie  forms  were  well  oiled.  In  order  to  insure 
plumbing  of  the  specimen  and  proper  alignment  of  the  hinges  in 
the  columns  of  the  test  specimens,  tlie  form  was  erected  in  the 
place  arranged  for  making  the  load  test,  with  the  steel  shoes 
and  hinge  pins  in  position.  The  reinforcing  bars  were  bent  as 
required  in  an  Olsen  cold-bend  testing  machine,  and  were  wired 


• O' 


i 


:..rc  .-6Xoo  I’V  :S.O  ff  'i  •;  L • '•,•■ 


44 


in  place  after  "being  set  in  the  form.  Concrete  \ms  dumped  from 
the  mixer  into  a tight  wooden  "box,  carried  to  the  form  "by  a 
traveling  crane,  and  shoveled  into  the  form*  A considera"ble 
amount  of  tamping  and  rapping  was  req_uired  to  get  the  concrete 
into  place,  especially  at  the  corners  of  the  frame*  The  forms 
were  stripped  when  the  concrete  was  about  24  hours  old  and  the 
specimen  was  lifted  off  of  the  test  "base  and  transferred 
to  another  place  in  the  la"boratory.  Wet  burlap  was  kept  around 
the  ^ ecimens  up  to  the  time  of  testing* 

8*  Testing  Apparatus*-  A heavy  concrete  test  base  was 
made  especially  for  this  investigation*  A general  view  of  the 
base  with  a specimen  in  position  for  testing  is  shorn  in  Fig* 21. 
It  ¥/as  22  ft*  long,  5 1/2  ft*  high  and  2 l/2  ft.  wide,  and  was 
reinforced  to  withstand  a reversal  both  of  vertical  loads  and 
horizontal  thrusts.  A vertical  steel  link  at  the  end  of  the 
base  allowed  for  a practically  frictionless  horizontal  move- 
ment at  the  bottom  of  one  leg  of  the  specimen  under  test*  A 
60-ton  hydraulic  jack  acting  against  this  link  was  arranged  to 
produce  a horizontal  reaction  through  the  axis  of  the  hinge 
and  either  to  maintain  a fixed  distance  between  the  two  hinges 
or  to  move  the  hinge  in  or  out  any  desired  amount*  Such  move- 
ment was  measured  by  means  of  a micrometer  bar  which  had  a range 
of  several  inches* 

Downward  loads  were  applied  on  the  top  of  all  specimens  at 
two  points  2 ft*  4 in.  on  each  side  of  midspan.  The  distance 
between  them  was  one- third  of  the  nominal  span  from  center  to 
center  of  hinges*  The  diagram  of  Fig*  22  shows  the  arrangement 


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71 


Fig.  21,  General  View  of  Test  Apparatus. 


N 


47 


for  applying  loads  and  reactions.  The  vertical  loads  were  pro- 
duced hy  two  100-ton  hydraulic  ;)acks  acting  downward  on  a 
steel  "box  girder  which  transmitted  the  pressure  through  a heavy 
knife-edge  casting  and  a roller  to  the  specimen.  Steel  plates 
' embedded  in  plaster  of  paris  were  used  to  distribute  the  bearing 
pressure  over  the  concrete.  The  upward  reaction  of  the  jacks 
was  exerted  against  built  up  steel  sections  connected  by  six 
tie  rods  to  other  steel  sections  beneath  the  test  base. 

Two  strain  gages  were  used  in  the  tests,  one  of  4-in,  gage 
length  being  used  for  measuring  deformations  of  concrete  and  one 
of  8-in.  gage  length  for  measuring  deformations  in  the  steel 
reinforcement,  A continuous  row  of  gage  lines  was  located  on 
the  reinforcement  along  the  length  of  the  outer  face  of  the 
specimen  and  a similar  row  was  laid  off  along  the  inner  face. 
Deformations  of  the  cnncrete  were  also  measured  on  several 
gage  lines  along  the  sides  of  the  brackets. 

Deflections  were  measured  at  points  one  foot  apart  on 
both  the  horizontal  member  and  the  two  columns,  A black  linen 
thread  was  stretched  at  constant  tension  between  points  at  the 
two  ends,  at  mid-depth  of  the  girder.  Similar  threads  were  hung 
as  j3umb  lines  along  the  sides  of  the  columns.  Movement  of  the 
^ecimen  with  reference  to  the  thread  was  observed  by  means  of 
paper  scales  pasted  to  small  mirrors  and  attached  to  the  speci- 
men, Readings  which  were  taken  by  lining  up  one  edge  of  the 
thread  with  its  reflection  in  the  mirror  could  be  duplicated 


within  .01  inch 


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48 


IV.  TEST  DATA  AJJJD  DISCUSSIOII  OF  RESULTS. 

9.  Procedure  and  Phenomena  of  Tests.-  In  general, 
downward  loads  were  applied  to  the  test  specimens  in  increments 
of  30  000  lb.,  with  the  hinges  at  the  bases  of  the  coltimns  held 
in  a stationary  position.  Zero  readings  were  taken  with  no 
dovmward  load  except  that  of  the  loading  rig  which  weighed  about 
3 000  lb*,  and  with  just  enough  horizontal  pressure  appliedat 
the  hinges  to  tighten  up  all  movable  parts  of  the  hinge  appara- 
tus. 

The  effect  of  a reversal  of  stress  was  obtained  in  the 
testing  of  each  specimen  in  the  following  mfinner.  After  readings 
of  deformation  and  deflection  had  been  taken  under  a load  of 
90  000  lb.  all  downward  load  was  released.  The  movable  hinged 
end  of  the  specimen  was  then  pushed  inward  an  amount  sufficient 
to  produce  maximum  deformation  readings  at  critical  sections  as 
great  as  those  observed  under  the  90  000  lb.  load.  Following 
readings  under  this  condition  of  loading  and  still  without  apply- 
ing any  vertical  loads,  the  horizontal  jack  was  swung  around  to 
act  on  the  inside  of  the  hinge  and  the  movable  end  of  the  column 
was  thrust  outward  until  stresses  were  again  produced  ?vhich  were 
comparable  to  those  observed  under  the  90  000  lb.  load.  The 
change  in  distance  between  hinges  was  measured  in  both  cases. 

With  the  horizontal  jack  swimg  back  to  the  outside  of  the  frame 
and  the  distance  between  hinges  brought  back  to  its  original 
amount,  vertical  loading  was  resumed  on  the  top  of  the  frame.  Com- 
plete readings  of  deformation  and  deflection  were  taken  at  a load 
of  120  000  lb.,  and  at  increments  of  30  000  lb.  up  to  the  maximum 
load.  Final  readings  were  t iken  in  each  case  after  the  maximum 


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49 

load  was  applied,  in  order  to  obtain  information  on  the  manner 
of  failure.  Fig.  23  to  27  show  views  of  the  different  speci- 
mens after  failure  had  taken  place.  The  following  paragraphs 
-give  a sliort  description  of  the  principal  phenomena  of  the  tests. 

Specimen  13A1»-  Loads  were  applied  as  indicated  in  the 
tabulated  data  of  Section  16.  Numerous  cracks  were  observed 
under  the  60  000  and  90  000  lb.  loads.  After  application  of 
end  thrusts  inward  and  outward  with  top  load  released,  the 
movable  hinge  did  not  return  entirely  to  its  original  position, 
but  was  pushed  back  into  place  mth  little  effort.  Failure 
occurred  at  a maximum  load  of  120  000  lb.,  with  noticeable 
crushing  and  spalling  at  the  north  inside  comer.  This  spalling 
was  apparently  due  largely  to  slipping  of  bars  at  inside  face  of 
column  near  corner.  There  was  also  apparent  slipping  of  tension 
bars  near  midspan.  Tension  cracks  were  numerous  in  middle 
portion  of  frame,  also  across  the  tee-flanges  on  both  vertical 
and  horizontal  members  near  both  corners. 

Specimen  13A2. - A number  of  tension  cracks  appeared  near 
midspan  at  the  30  000  lb.  load.  Cracks  also  appeared  across  the 
tee-flanges  about  15  inches  frcm  each  corner  on  both  horizontal 
and  vertical  members.  At  loads  of  60  000  lb.  and  90  000  several 
more  cracks  opened  in  same  regions.  V/ith  inv/ard  thrusts  several 
new  cracks  opened  in  the  top  face  within  the  middle  half  of  the 
girder.  With  outward  thrusts  tension  cracks  were  opened  at  the 
inside  corners  of  the  frame.  Failiii'e  occurred  through  crushing 
at  the  north  inside  corner  and  by  tension  in  the  steel  at  the 
outside  of  the  same  corner.  While  the  tension  failure  may  have 


50 


Fig.  23.  Views  of  Specimens  13A1  and  13A2  after  Test 


■I  , 


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53 


Pig.  27 


Views  of  Specimens  1302,  13D2,  and  13B1  after  Test 


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55 


occured  first,  the  yield  point  was  exceeded  in  the  compression 
reinforcement  at  the  comer,  and  also  in  the  tension  reinforce- 
ment under  the  load  points.  There  were  pronoimced  radial  cracks 
aronnd  both  corners  of  the  frame.  Maximian  load,  119  000  Ih. 

Spe cimen  12B1.-  failure  began  with  crushing  of  the  concrete 
at  the  north  inside  corner  at  intersection  of  bracket  and 
girder.  After  this  comer  had  yielded  somewhat  a large  nxunber 
of  diagonal  cracks  appeared  between  the  bracket  and  load  point. 
Cracks  were  not  large  at  other  parts  of  the  frame.  Horizontal 
cracks  in  the  tee-flange  in  the  north  end  of  girder  indicated 
that  flange  was  shearing  loose  from  the  web,  at  failure.  Large 
tension  cracks  were  observed  on  the  top  face  of  girder  at  the 
north  end.  Some  crushing  occurred  at  the  top  of  the  south 
bracket  and  at  the  bottom  of  the  north  bracket.  Maximum  load, 

15£  000  lb. 

Specimen  15B2.-  At  30  000  lb.  load  there  were  a few 
straight  tension  cracks  on  the  lov/er  side  of  girder.  With  a 
load  of  60  000  lb.  a number  of  cracks  opened  up  on  the  outside 
faces  of  the  legs  and  on  the  top  face  of  the  girder  near  the  ends, 
With  inward  thrusts,  a few  additional  cracks  appeared  in  the 
top  face  of  the  girder,  one  being  between  the  load  points.  With 
outward  thrusts,  a few  cracks  opened  at  the  junctions  of  girder 
and  brackets.  At  a load  of  120  000  lb,  piunounced  diagonal 
tension  cracks  developed  in  the  web  of  the  girder,  running  out- 
ward from  the  load  points  at  an  angle  of  about  45  degrees.  The 
maximum  load  was  reached  at  138  000  lb.  Failure  came  when  the 
yield  point  of  the  steel  was  reached  in  the  outside  face  of  the 
south  leg  a.nd  at  midspan.  At  about  the  same  time  crushing 


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56 


occurred  at  the  jimction  of  the  north  leg  and  "bracket,  and  the 
concrete  spalled  off  considera"bly.  There  was  a slight  indication 
of  crushing  at  the  south  end,  near  the  junction  of  leg  and  brack- 
et* 

Specimen  1501* - Several  cracks  opened  near  midspan  at  a 
load  of  30  000  lb.  At  60  000  lb.  load  cracks  opened  at  about 
midheight  of  the  outside  faces  of  both  legs.  With  inv/ard  thrusts 
cracks  opened  in  the  outside  faced  and  one  or  to'O  at  each  end  on 
the  top  of  girder.  With  outward  thrusts  cracks  opened  in  upper 
part  of  bracket  and  ran  do^'n  at  about  45  degrees  with  the 
horisontal,  parallel  to  the  face  of  the  bracket.  Cracks  of  this 
type  were  produced  in  all  frames  with  this  kind  of  loading,  and 
were  at  right  angles  to  those  produced  by  inward  thrusts.  At 
loads  of  120  000  and  150  000  lbs.  cracks  began  to  run  through 
from  the  outer  faces  of  the  legs  do\mward  diagonally  across  the 
webs  tov/ard  the  inside  faces  of  legs  at  junction  of  bracket  and 
leg.  At  the  maximum  lo  d of  182  000  lb.  crushing  occurred  at  the 
bottom  of  the  north  bracket.  Large  cracks  opened  in  the  outside 
face  opposite  the  crushed  area  and  the  yield  point  of  the  tension 
steel  was  passed  here  and  at  midspan. 

Sp ecimen  13G2. - A few  cracks  were  observed  near  midspan 
at  a load  of  30  000  lb.,  and  others  opened  up  across  the  outer 
face  of  both  legs  at  the  loads  of  60  000  and  90  000  lb.  Several 
cracks  opened  in  upper  face  of  girder  when  inward  thrusts  were 
applied  alone,  and  with  outward  tlirusts  the  cracks  were  similar 
to  those  found  in  Specimen  13C1,  At  a load  of  120  000  lb,  large 
cracks  opened  in  outside  faces  of  legs.  This  specimen  was 


57 


weakened  by  the  accidental  omission  of  two  of  the  four  longitudi- 
nal reinforcing  bars  in  the  outer  face  of  each  leg*  Due  to  this, 
failure  occurred  in  the  outside  face  of  the  south  leg,  opposite 
the  bottom  of  bracket  where  crushing  fc^ilure  rapidly  followed. 

The  maximum  load  was  148  000  lb. 

Specimen  IgPl. - A few  cr-^cks  were  observed  near  midspan  at 
the  load  of  30  000  lb*,  and  others  opened  up  across  the  outer 
faces  of  both  legs  at  loads  of  60  000  and  90  000  lb.  Several 
crocks  opened  in  the  upper  f .~ce  of  the  girder  i7hen  inward  thrusts 
were  applied,  and  outside  of  the  south  load  point.  At  later 
loads  this  crack  gave  the  impression  of  impending  diagonal  ten- 
sion failure,  however , failure  did  not  occur  in  this  part  of  the 
frame.  At  a load  of  180  000  lb,  large  cracks  appeared  at  both 
ends  at  the  top  of  the  veritcal  faces  of  the  legs.  The  maximum 
load  was  208  500.  Failure  occurred  simultaneously  when  the 
yield  point  of  the  reinforcement  was  re^^^ched  in  compression  on 
the  inside  face  and  tension  on  the  outvSide  face  of  the  south  leg 
at  about  mid-height.  The  concrete  crushed  over  a considerable 
area  in  the  locality  of  the  failure. 

Specimen  13 D2 , - A few  cracks  were  observed  on  the  outer  faces 
of  the  legs,  near  the  corners,  and  near  midspan  at  the  60  000  lb. 
load.  With  inward  thrusts  several  cracks  opened  on  the  top  face. 
With  outward  thrusts  cracks  on  the  tension  side  of  the 
girder  were  opened.  Large  cracks  appeared  under  the  180  000  lb. 
load  across  the  face  of  the  north  leg,  and  failure  occurred  by 
tension  in  the  reinforcement  of  the  north  leg  about  three  feet 
from  the  top  and  crushing  on  the  inside  of  the  leg  below  the 


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58 

jiinction  of  bracket  and  lower  haimch*  A ntunber  of  diagonal  cracks 
ran  between  the  sections  of  tension  and  crushing  failure. 

Specimen  15E1.-  In  this  specimen  erf  rectangular  cross- 
section  the  reinforcement  was  crowded  together  closely,  and  was 
probably  not  as  nearly  in  its  designed  position  as  in  the  other 
specimens*  At  a load  of  30  000  lb.  there  were  a large  number  of 
cracks  near  midspan,  around  the  corners  of  the  frame,  and  across 
the  outside  faces  of  the  legs.  At  60  000  lb*  load,  several 
diagonal  cracks  had  opened  betv;een  the  south  load  point  and  the 
end  of  girder.  At  a load  of  77  000  lb.  the  top  of  the  specimen 
began  to  crush  just  inside  the  south  load  point.  The  crushing 
extended  from  the  edge  of  the  bearing  plate  for  a distance  of 
several  inches.  With  continued  loading,  the  load  dropped  off 
somewhat.  Large  tension  cracks  formed  under  the  south  load 
point  and  at  the  south  corner.  It  was  noticeable  that  a large 
number  of  tension  cracks  developed  around  the  corners  of  the 
frame,  radiating  tov/ard  the  inside  corner  as  a center.  These 
cracks  extended  to  ?/ithin  3 to  3 1/2  inches  of  the  inside  or 
compression  face. 

Values  of  the  ultimate  loads  and  other  data  of  the  tests 
are  given  in  Table  4. 


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59 


TABLE  4. 

DATA  OP  TESTS. 


Specimen 

Ho. 

Age 

Days 

Ultimate  Loa 
poimds 

.d, 

Manner  of  Failure 

15A1 

43 

120 

000 

(Bond  at  outside  of  north  corner 
(followed  hy  crushing  at  inside  of 
(corner.  Tension  at  middle  between 
(load-points. 

13AE 

39 

119 

000 

(Compression  at  inside  of  north  corner. 
(Tension  at  outside  of  corner  and  at 
(middle  between  load  points. 

Av. 

41 

119 

500 

13B1 

42 

152 

000 

(Compression  at  top  of  north  bracket. 
(Some  shearing  between  tee-flange  and 
(web  at  north  corner. 

13B2 

40 

138 

000 

(Tension  at  middle  of  top  member  and  at 
(outside  of  north  comer  on  leg;  fol- 

Av. 

41 

145 

000 

(lowed  by  crushing  at  bottom  of  north 
(bracket.  Shearing  between  tee  and  web 
(at  corner. 

1301 

39 

182 

000 

(Compression  at  bottom  of  north  bracket 
(Tension  at  outside  of  hoc  th  bracket 
(and  in  top  member  between  load  points. 

1302 

41 

148 

000 

(Tension  at  south  end  in  outside  face 
(at  bottom  of  bracket,  followed  by 
(crushing  on  inside  at  bottom  of 

Av. 

40 

165 

000 

(bracket.  This  specimen  was  weakened 
(by  the  accidental  omission  of  2-1" 
(round  rods  in  outside  face  of  each 
(column  where  failure  occurred. 

13D1 

40 

208 

500 

(Compression  at  south  end  at  middle 
(of  lower  haunches  and  tension  in 
(outside  face  of  column  at  bottom  of 
(bracket. 

13D2 

41 

232 

000 

(Compression  at  north  end  at  bottom  of 
(lower  haunch,  tension  at  outside  face 

Av. 

41 

220 

250 

(at  bottom  of  bracket,  and  in  top 
(member  between  load  points. 

13E1 

41 

77 

000 

(Compression  just  inside  of  south  load 
(point. 

* T*  ■ ••"i**^*  *:  ' * i'' 


j ‘-  ' ’ ‘ ' ^ ••'  ',-ro;ij®--’  “:r?xo.X>  ”7"'  '■  ,•  ."■ 

’ ••  . . '>r  - •»«■  ■ •••)  -0.<Xv  XC? 

• 3'  " . - ■ ^ .i  wO'.  .;■  . *?'"'  -V  . , j. 


60 


10*  Thrusts  and  Moments.-  The  ratio  of  the  end  thrusts  to 
the  downward  load  for  the  tests  of  frames  with  hinges  held 
stationary  was  calculated  from  the  gage  readings  of  the  hydraulic 
jacks.  This  ratio  in  general  seemed  about  constant  for  each 
specimen  until  near  the  maximum  load  v;hen  it  decreased  consider- 
ably. This  could  probably  be  ascribed  to  a decrease  in  the  moment 
of  inertia  of  sections  at  points  of  high  stress.  Obviously  at 
stresses  near  the  ultimate  strength  less  elasticity  of  action  woul( 
be  expected  than  at  lower  loads. 

Table  5 presents  (a)  ratios  of  horizontal  thrust  to  total 
vertical  load  (average  for  all  but  ultimate  loads),  (b)  average 
actual  bending  moments  calculated  from  measured  vertical  loads 
and  horizontal  thrusts,  and  (c)  bending  moments  by  the  analytical 
method  described  in  Section  3 on  the  assumptions  that  the  moment 
of  inertia  varied  throughout  all  specimens,  first  as  the  cube 
of  the  depth  of  section,  and  second  as  the  5/2  power  of  the  depth 
of  section.  A very  fair  agreement  betv/een  calculated  and  ob- 
served moments  is  seen. 

Values  of  the  bending  moment  at  midspan,  taken  from 
Table  5 are  plotted  in  Fig.  28  as  ordinates  against  horizontal 
lengths  of  bracket  as  abscissas.  Since  the  frames  of  Type  D 
had  a haunch  different  in  shape  from  those  of  the  other  frames 
it  was  found  convenient  to  reduce  it  to  an  equivalent  length  of 
45  degree  bracket.  From  calculations  it  was  found  that  the  haunch 
of  Type  D would  produce  the  same  moment  at  midspan  as  a 45  degree 
bracket  32  inches  in  horizontal  length.  The  points  in  Fig.  28 
which  represent  actual  bending  moments  as  determined  by  test  are 


;0  9LUJ9J,  U/  uods'p/l^  J.U9LUOl^ 


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Fig.  28.  Calculated  and  Observed  Moments  at  Midspan 


62 

seen  to  agree  fairly  well  with  the  calculated  bending  moments 
based  on  Equation  1.  Considering  the  variable  nature  of  the 
materials,  the  difference  in  details  of  design  and  dimensions, 
and  the  various  possible  sources  oferror  in  observations,  this 
might  be  expected.  Eurthermore,  it  indicates  that  the  analysis 
of  the  elastic  homogeneous  rigid  frame  may  be  applied  without 
great  error  to  the  structure  of  reinforced  concrete. 

TABLE  5. 

COMPARISON  OP  OBSERVED  AND  CALCULATED  MOMENTS 


Specimen 

No. 

H 

Average 

of 

Test 

Ratio 

of  Bending 

Moment  to 

6 

Prom  Test 

• Calculated  f I=:kd^  i Calculated  (L=M^' 

At 

Mid- span  corner 

• At  At 

: Mi  d-€pen  Corner 

: At  At 

: Mid -span  Comer 

• 

13A1 

.168 

.568 

.432 

15A2 

.202 

.480 

.520 

Av. 

.185 

.524 

.476 

.481 

.519 

.481 

.519 

13B1 

.200 

.486 

.426 

13B2 

.236 

.392 

. 520 

Av. 

721F 

.439 

.47^' 

.370 

.524 

.379 

.517 

le3Cl 

.241 

.438 

13C2 

.305 

.602 

Av. 

.273 

.520 

.258 

.536 

.271 

.526 

13D1 

.312 

.198 

7FT4“ 

13D2 

.352 

.095 

.674 

Av. 

.332 

.146 

.624 

.186 

.590 

.203 

.577 

13E1 

.218 

.440 

.560 

.481 

.519 

.481 

.519 

The  variation  of  bending  moment  throughout  the  frames  may  be 
seen  in  Pig.  29  v/hich  shows  graphically  the  relative  magnitude 
of  the  actual  bending  moments  and  their  distribution  in  the 
different  types  of  frame,  as  determined  by  the  tests.  This 
variation  is  due  to  two  distinct  factors,  variation  in  stiffness 


63 


/?£rLf^T/V£: 

/// 

D£r£Rn/N£:o  by  Tf:sr.5 


Pig,  29,  Relative  Moments  in  Frames, 


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64 


and  variation  in  the  shape  of  the  axis  of  the  frame,  as  noted 
in  Section  3*  It  is  seen  that  while  the  moment  at  mid-span 
vafies  with  the  size  of  the  "brackets,  the  moment  at  the  corner 
of  the  frame  is  about  the  same  in  all  types  of  frame. 

11.  Flexural  Stresses  and  Deformations.-  Ta"bulated  strain 
gage  data  from  the  tests  of  all  frames  are  given  in  Section  16, 
which  also  contains  diagrams  showing  the  position  of  all  gage 
lines  for  measurement  of  deformations  and  deflections,  and 
shows  the  position  with  respect  to  these  ga^ge  lines  of  cracks 
and  crushed  areas  observed  at  the  maximum  loads. 

Unit  stresses  in  the  steel  reinforcing  bars  as  determined 
from  strain  gage  measurements  are  sho?m  graphically  in  Fig.  30 
to  30.  lYhile  the  stresses  sh avn  represent  combined  flexural 
and  direct  stresses,  the  latter  (which  were  compression  for  all 
cases  of  downward  loading)  were  comparatively  small  and  did  not 
exceed  7 per  cent  of  the  total  maximum  stress  in  the  extreme  case 
of  the  frames  of  Type  D.  For  the  other  frames  the  direct  stress- 
es are  much  less  and  hence  do  not  have  an  appreciable  influence 
upon  the  total  stresses  measured. 

The  calculation  of  stresses  in  concrete  and  steel  of  these 
frames  is  quite  laborious,  due  to  the  many  changes  in  cross- 
section,  reinforcement,  and  the  variation  in  the  bending  moment. 

A sufficient  nmber  of  calculations  have  been  made,  hov/ever,  to 
show  a fair  agreement  with  the  resrkts  of  the  tests.  An 
example  of  the  comparison  between  calculated  and  observed  stress- 
es is  shown  in  Table  6,  which  gives  stresses  in  the  tension 
reinforcement  at  midspan  due  to  a number  of  different  loads. 


» ifj' 7r:- ui.  .'4.. 


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66 


ai- 


nner 


.‘^O'hc.5-  , 

w'ere  m<zas>ur<zd  on  rizinforoincj  bar£> 

Tknsiion  p!off<zd  mv/ard  on  Jzff  ha/f  of  fram<z- 
and  oufward  on  n^hf  half  of  fram(Z^ 


30  20  to  O !0  20  30 
3fr(zsiSr  dhous^and  lb  per  odj  m 


Specimen  No  /3A-I 


30  ZO  to  o 
fj  fre  S3 -thous^and 


!0  20  30 
lb  per  3 f 


- IZOOOO 

--  >70000 

- ^oooo 

■■  30000 


m 


Fig,  50.  Observed  Stresses  in  Specimen  13A1. 


Fig,  31.  Observed  Stresses  in  Specimen  13A2 


66 


t Noi<z.s>  . 

AH  3fr^5>s>e.s>  me.a^ur<z.d  on  nzinforcinc^ 

Thns>':on  piofiod  mw^rd  on  left  half  of  fran 
- — and  out yyard  on  nc^ht  half  of  frames 

vr  , Specimen  No I3B- 1 a 

30  ZO  to  O !0  ZO  3>0 


3frzF^B>-fhous>nnd  !t>.  per  5-^.  m. 


!0  ZO  30 

3lrcs>s>-fhou3>and  !b  per  Sxj.  m 


Fig.  32.  Olsserved  Stresses  in  Specimen  13B1, 


jirf-*''"  V®.  “ 


67 


Fig,  34,  Observed  Stresses  in  Specimen  13Cl, 


Fig,  35,  Observed  Stresses  in  Specimen  1302 


68 


>o 

P=  J 50000 

'P-  /zoooo 

P‘  90000 
&0000 
30000 


30  ZO  !(}  O /O  ZD  30 
S>'/'re:5>£>-thous>iOJ7c/  /i>  p/zrsx^./r?. 


Specimen  No  /3D-I 


30  ZO  to  O 
5>lrss>3>  -ihous>af7c/ 


O ZO  30 

/b.  per  5^  /n 


Fig,  36,  Observed  Stresses  in  Specimen  13D1, 


P- 

Z.'0.'’0C 

■ p = 

50000 

P’ 

/zoooo 

-p. 

9rc~-0 

p ^ 

iOOC' 

300CC 

J--  j 

O to  ZD  30 
iran^  J lb.  Dzr  3^  m 


Specimen  No  /3D-. 


, o O >0  Z 
lm,Z-  -thouo  tn  J 't,  or- 


Pig,  37,  Observed  Stresses  in  Specimen  13D2, 


69 


Pig,  38,  Observed  Stresses  in  Specimen  13E1, 


70 


The  observed  stresses  were  determined  from  the  average  deforma- 
tions measured  on  gage  lines  22  and  22a  of  each  specimen,  while 
the  calculated  stresses  are  based  on  bending  moments  computed 
from  the  known  loads  and  reactions  on  the  frames,  and  include 
the  small  compressive  direct  stresses  which  existed  at  the  sec- 
tion considered.  The  calculations  ?i^ere  made  on  the  conventional 
"Straight-line  theory"  of  stress  distribution  and  follow  the 
assumption  that  no  tension  is  carried  by  the  concrete.  In  most 
cases  the  table  shoves  the  calculated  stress  in  the  reinforcement 
to  be  higher  than  tho.t  found  from  test,  thus  indicating  that  seme 
part  of  the  tensile  stress  was  carried  by  the  concrete. 

TABLE  6. 

COUPARISOH  OF  CALCULATED  AED  OBSERVE!)  TENSILE  STRESSES 

IN  RSniFORCEIvENT. 


Specimen  Stress 

Load  on  Specimen  - lb. 

No. 

'30  000  60  000  90  000  l20  000 

Maximum 

Load 


13A1 

Calculated 

11 

500 

32 

000 

42 

300 

61  400 

61  400 

Observed 

12 

000 

21 

800 

31 

600 

Yield  Point 

Yield  Point 

13B1 

C. 

10 

000 

22 

000 

34 

500 

55  90b 

60  400 

0. 

7 

100 

19 

600 

30 

000 

40  100 

44  500 

13C1 

C. 

7 

100 

13 

000 

23 

600 

45  500 

46  500 

0. 

8 

600 

13 

700 

21 

700 

31  000 

Yield  Point 

13D1 

c. 

5 

300 

11 

700 

12 

900 

18  500 

50  000 

0 

7 

500 

11 

000 

18 

200 

20  200 

58  200* 

13E1 

C. 

8 

600 

32 

400 

42  900 

0. 

18 

100 

34 

900 

47  800 

* Yield  Point  on  steel  - 55  000  to  60  000  lb.  per  sq.  in. 


As  an  aid  to  further  comparison  between  observed  and  calculated 
stresses,  Fig.  39  shows  in  a general  way  the  relative  magnitude 
and  distribution  of  the  tensile  stresses  in  the  steel  and  the  com- 
pressii©  stresses  in  the  concrete  for  all  frames  under  any  given 


If  ■ '■  . ' ■ ■-••,■  --r  ■ 

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71 


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A 


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P/df^ra/T73  3hoyy//7^  re/^:/'/i^(^ 
3ir(Z3Se:3  on  c/iffer^n:/-  J-pp<^5 
o:f  5ubr<zc:f<z’c/  :/o  e^oa/ 

y<zr//a^/  /o^^as.  £3  <^rs5an7e:c/ 
■/■a  in^rp  03  ^ £c  io 
as  7^  £>roken  l/n<z5  Jnc//caJ‘<s^ 
r(^/a//y<z  honc/zne^  /Tycm^n:^ 


Fig.  39.  Relative  Stress  Distri^btition  in  Frames. 


7E 

loading.  This  figure  is  merely  an  approximation,  hut  the  shapes 
of  the  stress  diagrams  compare  very  well  with  those  shown  in 
Fig.  30  to  38. 

When  the  radius  of  a curved  member  is  small  as  compared  to 
the  depth  of  the  member,  the  well-known  theory  of  curved  beams* 
indicates  that  a linear  distribution  of  stress  across  the  section 
of  the  member  does  not  exist,  but  that  the  stress  on  the  concave 
face  is  greater  than  it  would  be  in  a similar-sized  straight 
member  subjected  to  the  same  moment.  To  investigate  this  feat- 
ure of  analysis,  especially  for  specimens  of  Type  A and  E which 
have  only  S-inch  fillets  at  the  inside  corners,  strain  gage 
readings  were  taken  on  the  concrete  on  the  side  of  the  web  at 
one  corner  of  all  specimens.  Using  the  measured  deformations 
of  the  concrete  and  the  deformations  in  reinforcing  bars  in 
the  tension  and  compression  faces  of  the  specimens  at  the  corners, 
the  position  of  the  neutral  axis  for  each  specimen  was  determined, 
as  closely  as  possible  and  is  plotted  in  Pig.  40.  These  de- 
terminations were  not  exact  because  of  the  small  number  of  gage 
lines,  and  from  the  fact  that  all  lines  were  not  located  to  the 
best  advantage.  The  deformations  used  were  chosen  at  loads 
producing  tensile  stress  in  the  reinforcement  of  from  20  000  to 
30  000  lb.  per  sq.  in.  (^Deformations  which  were  observed  at  the 
60  000  lb.  load  exceed  the  deformations  calculated  as  in  a 
straight  beam  by  20  per  cent  on  Specimen  13A2  and  by  65  per  cent 
for  Specimen  13E1.  It  is  evident  that  this  high  concentration 
of  fibre  stress  should  be  reduced  and  hence  it  would  seem  ad- 

*See,for  instance.  Strength  of  Materials, p. 323,  by  J.E.Boyd. 


(jQ 


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73 


Po^/rtoA^s  or  Nei/rj^/iL  Px£rs 

v5/ee/  >5tre33  =•  2*0000  /o  30,000  y/z  /r?. 


Pig,  40*  Neutral  Axes  in  Curved  Beams, 


74 


visalDle  that  some  sort  of  fillet  or  bracket  should  always  be  used 
in  a sharp  corner  of  this  sort*  The  minimum  size  of  bracket  to 
produce  a satisfactory  stress  distribution  is  problematic,  but 
the  horizontal  length  of  the  bracket  should  probably  be  not  less 
than  one-half  the  depth  of  the  adjoining  members,  and  preferably 
greater*  A curved  fillet  or  bracket  should  produce  the  best  vari- 
ation of  stress  around  the  corner* 

IE*  Shearing  Stresses. - In  all  cases  the  frames  were  highly 
reinforced  against  diagonal  tension  by  the  use  of  U-stirrups  and 
bent-up  longitudinal  bars.  The  ends  of  stirrups  were  hooked, being 
bent  out  into  the  flange  of  the  T- sections  and  bent  inimrd  in  the 
rectangular  sections*  The  bent-up  bars  were  also  anchored  by  semi- 
circular hooks  having  a radius  of  four  diameters  of  bar. 

The  effectiveness  of  the  web  reinforcement  was  demonstrated  by 
the  fact  that  none  of  the  frames  failed  by  diagonal  tension,  and 
that  diagonal  cracks  were  in  general  guite  small*  Knowing  this, 
the  shearing  unit  stresses  which  were  developed  seem  q.uite  note- 
worthy, inasmuch  as  they  are  much  higher  tlian  have  been  found  in 
any  tests  outside  the  investigations  of  the  Concrete  Ship  Section. 

Shear  diagrams  for  the  different  frames  at  maximum  load  are 
given  in  Pig*  41,  and  Table  7 gives  values  of  the  unit  shearing 
stresses  developed,  as  calculated  at  sections  of  maximum  shear 
just  outside  the  load  points.  Specimens  13A1  \nd  13D2  inclusive 
were  of  T-section,  having  a flange  30  inches  wide  and  3 inches 
thick,  and  the  value  of  j used  was  0.88.  Specimen  13E1  was  of 
rectangular  section,  and  the  value  j used  was  0.83,  A comparison 
is  also  made  in  the  table  between  the  shearing  unit  stress,  and  the 
compressive  unit  stress  of  the  concrete  as  determined  from  tests 


\^=S3800 


T^pe  A 


r=SZ50O 


Type  C 


?=33SOO 


7Zf-00 


Tf^peE 


E=//a/oo 


Fig.  41,  Shear  Diagrams  for  Frames 


at  Maximtira  Loads 


76 

of  6 X 12  in*  cylinders  made  and  tested  vath  the  frames, 

TABLE  7. 

mxiimi  OBSERVED  SHEARING  STRESSES  IN  FRAMES 


(No  fail-ore  occ-urred  through  diagonal  tension)  • 


Speci- 
men No 

Maxim-um 
. Vertical 
Shear 
lb. 

Width  of  Depth  of 
Section  Section 
in.  in. 

1 

in.  ^ 
lb. 

Shearing 

Unit 

Stress 

V lb 

per  S(i.in, 

Cylinder  Ratio 
Strength  v 

fc  '^0 

.per  sq.in. 

15A1 

60  000 

8,12 

9.75 

6 9.7 

860 

3110 

.28 

15A2 

59  500 

8,00 

9,75 

6 8. 6 

865 

3765 

,23 

15B1 

76  000 

8.12 

10.0 

71.5 

1060 

4260 

.25 

13B2 

6 9 000 

8.00 

9,75 

68.6 

1005 

3665 

.27 

1301 

91  000 

8.25 

9,75 

70.8 

1285 

4505 

.29 

1502 

74  000 

8.27 

9,75 

71.8 

1030 

4040 

.26 

13D1 

104  200 

8.12 

9,87 

70.6 

1475 

4420 

.33 

13D2 

116  000 

8.12 

10,12 

72.3 

1605 

5860 

.27 

13  El 

38  500 

8.00 

8,92 

59.2 

650 

4995 

.13 

It  will  he  noted  that  the  shearing  stress  of  650  Ih,  per  sc[, 
in,  for  th  rectangular  section  is  about  as  high  in  comparison 
to  the  strength  of  the  concrete  as  the  highest  values  found  in 
previous  test  of  ordinary  beams,  while  the  values  for  th©  T-beaia 
sections  are  much  higher.  It  is  true  that  shearing  stresses  were 
accompanied  by  a small  direct  compression  which  balanced  a little 
of  the  stress  in  the  tension  side  of  frames,  and  may  have  reduced 
somewhat  the  tendency  to  diagonal  tension  failure;  still  there  were 
generally  fine  vertical  tension  crachs  present  just  outside  the 
load  points  at  very  low  loads.  It  does  not  seem  likely  that  the 


77 

direct  compression  produced  any  consideralDle  increase  in  the 
resistance  to  diagonal  tension. 

Due  to  the  fact  that  the  positive  and  negative  moments  in 
continuous  frames  may  he  equalized  hy  the  ^‘udicious  use  of 
haunches,  the  magnitude  of  the  moments  is  kept  comparatively 
small;  conversely,  in  such  frames  the  shearing  stresses  x¥ill 
be  correspondin^y  large.  Hence  it  is  of  considerable  value 
to  find  that  safe  shearing  strengths  may  be  obtained  which  are 
much  greater  than  these  caamonly  allowed  in  building  practice. 
This  is  clearly  dependent,  however,  upon  the  use  of  a suffi- 
cient amount  of  web  reinforcement,  properly  distributed  and 
anchored,  and  upon  proper  anchorage  of  the  longitudinal  rein- 
forcement. 

13.  Moment  of  inertia. - It  is  well  known  that  statically 
indeterminate  bending  stresses  are  governed  by  the  relative 
stiffness  of  the  various  parts  of  a structure.  The  stiffness 
of  a member  in  flexure  is  iBually  measured  by  two  quantities;  1, 
the  moment  of  inertia,  which  is  a function  of  the  size  and 
shape  of  the  cross  section,  and  E,  the  modulus  of  elasticity, 
which  is  a physical  property  of  the  material.  Ho¥/ever,  in  a 
composite  member  of  steel  and  concrete,  the  latter  of  which  is 
so  deficient  in  tensile  strength,  both  E and  vary  with  the 
stress  in  the  member.  A very  large  reduction  in  1 occurs  when 
the  concrete  fails  on  the  tension  side  of  the  member,  and  a 
further  reduction  takes  place  in  E and  _I  as  the  concrete  fails 
to  take  compressive  stress  in  proportion  to  deformation.  Through- 
out this  variation  in  stress  distribution  for  the  concrete  part 


'v;i.r*  Mr.  ■-^tr,. 


1 


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t • ’ 

.*  .'^•".  . b ‘‘l  - • ' -' 

- >r  ij, , , ' 'ifif^! 


: *. 


r C'. 


..  » 


..  1. 


‘i  . '■  . /•  1o  kJ 

) • ; 

‘ o ■ ' 

• : tcooir  .c*' 


0.-  •,  :;v.' 


V!  '■ : 


;.'*>»  1 


I-' 


;..0  piij 


,4  ■ - ' , " 


■ '.)  r 


rr  't  jj*.'*  J 


ftc  • •'i  'i';  •'.' 


* /*: 


r #.  ; yA.  . 

v-k.  '-C-'  , 


!1 


:.0  ^?'r 


: 


c:*.r  i': 


!.cX.  VO  , j.'._!A 


I 


,:,i  X;>  I wX  "'“o-.-co  |ij, 


.- V 


-t 


f 

I .. 


;'.  r>  o > 

K ' 

■ r^i'JC'Av  o '.'• ' a:  ■ •''  ■ 


' C’O  0 > > OA  t)  O * ‘.u  uv  7f  ^ 

" ^ 


O'j  I,  : ::;  .*i;!  , f)  / 


■ o.;J-  '• 


f ■- 


'.  .'.i» 


f.’sr 


V- 


• S *' 


..•■•0.  .ivio::  or- 

..  L,  ii/o  , S 
•i! 

• •:  '-  : .•  ■ J-  • 


78 


of  the  member  the  neutral  axis  chr^nges  and  this  in  turn  affects 
the  moment  of  inertia  of  the  steel  area  slightly. 

Aside  from  the  question  of  the  effect  of  stress,  the 
variation  in  the  moment  of  inertia  v;ith  the  shape  of  a member 
must  be  considered.  For  rectangular  areas  of  width  ^ , snd 
depth  d,  ^ varies  as  bd^.  If  such  areas  contain  equal  per- 
centages of  steel  similarly  placed,  1 still  varies  as  bd^. 

If  such  areas  contain  equal  areas.  A,  of  steel  similarly  placed, 
1 for  the  steel  varies  as  For  a tee  beam,  1 varies  as 

a 

bd  if  the  width  of  tee  and  the  ratio  of  flange  thiclmess  to 

p 

depth  remain  constant  and  as  something  more  than  if  the 
flange  thickness  remains  constant.  Hence  it  is  seen  that  I_ 
for  the  frames  tested  v/ill  vary  as  some  power  of  d between  d^ 
and  d^. 

In  calculating  _I  for  the  section  of  a reinforced  concrete 
beam,  especially  when  combined  flexure  and  direct  stress  are 
encountered,  a clear  distinction  must  be  made  between  center  of 
gravity  and  neutral  axis  of  section.  Consider  the  reaction  H 
acting  at  the  hinge  of  a frame  as  shown  in  Fig.  42.  Imagine 
the  portion  of  the  frame  ©ut  away  below  the  section  A-A,  leaving 
the  reaction  R,  acting  as  before.  Assiuning  that  the  concrete 
may  be  cracked  over  some  portion  of  the  tension  side  of  the 
member,  let  the  effective  area  of  the  section  be  the  sum  of  the 
uncracked  area  plus  an  imaginary  concrete  area  which  will  carry 
a stress  equivalent  to  that  in  the  reinf orceiment.  The  point 
a represents  the  centroid  of  this  effective  area.  By  introduc- 


ri- 


t ?t:.  r-.os 


— » . ^ V J J* 


i frr 


• ...  ‘ ’*'  f '. 

■^  ' ! -i  ■■•  fOM  y^T  .') 

' '"lli  «■  .. 

r !•  I *.;ivi?^  « ;i  V 

f-  •■ 


i ■ , ',  -i.  .'  .: 

■ , ' f 


•V  - 


- oi  ' Cl 


o'ir,.nr 


DP.:  -i;.'. 

i-e  J-  ro  al-:*--  . II  ^ 

. • VJ  ^.'  ''^V  .• 


V * 


'C  I ii  z( 


•J . 


) ■>■  ■> 


';  iOX  i\  "t, 

- , • .‘j- 


'V  J:yrzXr*  J-. 


■ .•epyJ'fcC  .•  "-/>  oo 

' • < .■  or-  -,*■  'ii-xi  ;>  , , ,- 

,.  .w 

» --  ' '.i>Oi*'.  V \aJliiLi 

O^v'L-..  r"  . Ii-  .^  '.' 

«i  'fi  '.":'}  u ' I I i"’  ' .■  ■ ',' ' '-• 


\ , 


-■  ■ ' ; i .r  ■ ; 

'■>'  . .'  ’ -i'.Vj  : 

- ' •:  * .-  ’ 

■•-  .■■  ^ ' :.vrt' 

.'  I ir  ■<  Ir  T ,.,t 

V-.' 

',  %)  u ■ 

,■*;  I : ■ f)  ■ ■:  ' »u( 


■•  :.'tr 


-'  , 'j 


iirl 


, , -r  ; :■ 

i cO  r -t:;>C|  -*v  ^ '•i  :*t 


t 


^ ! : r.jo'v  •;■•.:]■ 

' ’ ' - Til 

» 


.O'-'  ■'  ' ’ ■•►■’■ 

- ^ 


\ 


Ui 


t-  \'l.  ^ 


I2l  ® ^ 

• ' ■ ■'■  X/‘  V- 


4Jvl  >,.:^  o?(:t 

■-■1 

.H 

. " ?^l}v 


' J , rrr  ;j  r-;  ,■A;^a  > ..•^, 


..r>  ^ 

WN!*! 


, 4'  o /i  J 



, ■>  -V  ■ 


> i 

\i~4 


79 


Fig,  42,  Mera"ber  under  Combined  Flexure  and  Direct  Stress. 


80 


ing  a system  of  three  balanced  forces,  Rj  (equal  but  opposite 
to  R, } V,  and  H,  all  acting  at  a,  the  reaction  R is  resolved 
into  a shearing  force  H,  an  axial  force  V ^ couple  R,e« 

The  couple  produces  flexural  stresses  varying  from  f^  to 
fg^  being  zero  at  the  point  a.  The  axial  force  V produces  a 
uniform  compressive  stress  V over  the  effective  area,  so  that 
the  resulting  neutral  axis  due  to  combined  flexure  and  c express- 
ion is  at  the  point  b, 

Pran  the  foregoing  it  follows  that  the  center  of  gravity 
of  the  effective  area  is  to  be  used  in  finding  the  magnitude 
of  the  bending  moment  and  that  it  lies  on  the  axis  about  which 
the  moment  of  inertia  is  to  be  found.  The  neutral  axis  is  to 
be  used  in  determining  how  much  tension  may  exist  in  the  con- 
crete at  the  section. 

As  a method  of  determining  the  value  of  the  product  M 
from  test  data,  use  was  made  of  the  well-known  flOJ^ure  formula, 

M = ~ . This  may  be  written  in  the  form  El  = ^ in  which 
e_  is  unit-deformation  and  c_  is  the  distance  from  the  neutral 
axis  to  the  point  where  _e  is  measured.  M,  £,  and  e_  were  ob- 
tained from  test  data.  For  convenience  ^ has  been  considered 
as  having  a constant  value  of  3,750,000  lb.  per  sq.  in.,  which 
is  the  initial  modulus  of  elasticity  for  this  concrete,  and  all 
variation  in  the  quantity  M is  taken  care  of  in  the  single 
quantity  I.  Pig.  43  shows  the  variation  in  I at  different 
values  of  the  compressive  stress  in  concrete  as  determined  from 
sections  approximately  10.5  inches  in  effective  depth.*  The 

A somewhat  similar  variation  in  the  moment  of  inertia  of  a 
reinforced  concrete  beam  is  described  by  Dr. P. von  Emperger  in  an 
article  "Die  Wahre  Grosze  des  Tragheitsmoments  im  Eisenbeton- 
balken. " Beton  und  Eisen,  June  5,1916. 


■Ji  : 


■'  :,f ''-a 

^ A*  7/-I 


, * » 


r<i.:fr#a  .:  .r>;>:  'i.'cr 

■■■■  ■- 

■ *•  ) ' - V . ■ * ' X t ^ ■' 


'■'^i  'h'^ 


o;r’  ^\:. 


/ • * 


■,.r  '.  ■ ^ • '>♦•♦,' 


\' 


C»' /« .. 

?»’XfC':©r:v  J. 


/“'■  V t.-.-jj- 


'-r  : 


" ’A' '»  ’ p. 


'■  r.a 


T ..J;'-*i»  { 


~ .'  'rrvL 


. ^ii'W  '• 


. 1 * ^M  ist  fi 


r 


ft- x/o-iM  ' : :iJi% 


4v^j  i' 


X 


: ' ^ > ;t  ’ tu  !. 


.'  . ;it>cr  .'  ^v^  lo 


' 'V,s 


'■  '-  'Vvi'  a.-v!  *<•.  ./•».[.•  •'  0.'  ‘ 


- «(J 

■■.>*53 


5^:'  Kii  iv'; 


<<!'•  .-■. 

' JCOV.'i 


. ‘.^i.  ' . . i.-  . 'i 

> 


‘ • ' .;  itfi'-f ) i ■>,•*'  • »'  . : j: 

X ••; o i.t  ..:.>  j ';!  ,-,„  ^,1 

•ivj  ' , , ■)  . T 

ti  :fTv  iir  -i  i*  r , ' ’ ;,  i 


;V 


•v' 


■ ■ ' ' < •';■  > “ ."  42  ''■  i'i  ■ ‘ ‘ 

■■“'  i»fe' 


, V 

•j/f  j ■» . 


n . ,nf  / 


, ' , f;  j r?  O kf  iii'y  Jjyfc i-f 


,'.  >v',.  . 


'I'XO 


/ a 

iu.. 


• f ■..  f i 
^..:  ■ '^ 't  .cl..;- 

. -OH  . 

' 9*.,  . 


' j /■;■  <■  v>.  'Cii' 

■ _.,  ;v;  Vfl  •; 


•A^.* 


V\N 


>,  * .-  ‘i  ‘ 


’ f-  ;'*■  ^ , , ,»■'  • ; 


":  V ,jP  ’ ’ .*i'CT'i*J 

I • •>  O'T;.  ' V.V 


( • J.  '. 


V . ■ 1 . 


•'  -’  a ' ' t i f*i.^r.xv  £ic  f m m - J .7  '■ ' ■ ■ ,i  j‘  1 .'iCr !*'■'.  > ..i.'xi) 

\ ^ "''‘T' •r'JEs’t:  ;■  ."'.T 


Y .: 


Moment  of  /nert/o/~ 


81 


I 

H 


Fig,  43.  Variation  in  with  Compressive  Stress. 


wide  divergence  of  points  shown  may  he  sttrihuted  to  errors  of 
observation  in  M,  c and  e,  to  a considerable  variation  from  the 
nominal  depth  of  10,5  inches,  to  the  variation  from  the  design 
dimensions  of  the  section,  and  to  the  difference  in  the  steel 
area  used  in  different  specimens. 

Moments  of  inertia  calculated  in  the  ordinary  way  frcm  the 
nominal  dimensions  of  the  cross-section  of  the  test  specimens 
are  also  shown  in  Fig,  43.  A section  10  l/E  inches  in  effective 
depth  and  having  4,75  per  cent  of  steel  in  both  tension  and 
compression  has  been  used.  Using  a modulus  of  rupture  of  450  lb. 
per  sc[,  in.  for  this  concrete  and  a value  of  n equal  to  8,  the 
effective  section  at  different  stages  of  loading  has  been  de- 
termined. Values  of  1 computed  at  the  different  loads  have  been 
plotted  in  Fig,  43  against  values  of  the  calculated  compressive 
stress. 

A general  agreement  bet?/een  the  calculated  curves  and  the 
experimental  points  is  seen,  although  the  latter  are  quite 
scattering.  It  is  seen  that  until  the  concrete  in  the  tension 
surface  of  the  member  began  to  fail,  remained  constant,  and 
was  equal  to  about  3450  in,^.  The  tension  failure  of  the  con- 
crete began  when  the  compressive  stress  was  about  500  lb. 
per  sq,  in.,  and  after  this  the  moment  of  inertia  decreased 
rapidly.  The  average  values  of  the  experimental  data  are  repre- 


that  a similar  variation  in  moment  of  inertia  occurs  with 
with  different  depths  of  member.  Fig.  44  has  been  drawn.  Logar- 
ithms of  moment  of  inertia  and  depth  of  member  were  plotted. 


sented  roughly  by  the  hyperbola,  I = 3450 


To  show 


1 


' ^ 


j 0 


f; 


i 


o: 


J 


.'  ■* 

? . • • . 


i 


( 

5 

» 

i 

! 

* 

I 

I 


Zc?qafr/y/7/773  of  ffomen/3  of  /ner^/a. 


t 


•s 


oe  0.9  1.0  A/  1.2  AJ  A 4 

Lo^Ofr/Z/jms  of  Dep//?5  of  .5ecf /or?,  cf. 


Cu/?yE5  J/iOm/^6  ffEL/rr/ON  OE  LOO.  I TO  L06.d 

6or?errf/  ^r^ooLf/o/?  ^f? 3} . I =fbr?5fcnr?f  OeJoiLi/  ^ =500an<f  ./* 962^^ 

Fig.  44.  Variation  in  1 with  Depth  of  Meraher. 


v<  1 


■ 


' V' ; 


k; 


-^i  • ’ ' V - 'i^-' ■ ' ■ ■ f*' ' ' '■  ■» 


'v ' 


M ■■ 


V.' ■ IV  ■'•(*? 

■„  ■■  : >'!• 


;f«  >-  ii  aB  ' 


* • >-  --r 

’rr'gri 


Id 


■ '•■^i'’'®’’^'^:  -*±.:  . •'  •5; 

' ■■■;■'■  ' '‘  -(i 

* - ‘’W  , ‘ 

K,  S . * .;  .'.■  ■ V ''  'V  V^''--’'  ''^v'’"’v‘'"' 

■ ■•  i-v.ftftSaf /v.-'b 


• * J ■ . '*  ^ 


»*i<  'rn..  . 


}■/•,  '.r:>.v-r 


‘■Jil?*!' 'r.l! 


b-  . '■;  ',.■  « ’■  i.sai 

• , ■ »:*>  •.?  .v.^4,  i”  ■'  •,« 


■0 


>♦ 


V-*  ■•■  • ■ •'  • ' --iA^  '■■■  -*'i 

- *■■"'■'  '•■  V.yv  -•  r -^.digSI '•' 

«Hk,v,  ..  ...  ^ mMsm  ^ 


■ii 


t.'  . .Jf:, .,  ■’.i' . ■ .iif 


4». 


.i  ..  A ,'  iv  ‘ 


84 


taking  the  points  along  each  curve  from  data  obtained  within  a 
certain  range  of  compressive  stress  in  the  concrete.  From  the 
slopes  of  these  average  curves  _!  is  found  to  vary  approximately 
as  d Hence  each  curve  represents  an  ey^uation  of  the  form, 

I = kdV^.  The  values  of  k from  Fig.  44  are  found  to  decrease 
as  the  values  of  fo  become  larger,  in  the  same  general  way  as 
was  shown  in  Fig.  43. 

A general  expression  for  the  moment  of  inertia  of  sections 
of  the  test  specimens  is  found  from  the  data  of  Fig.  44  to  be 

I = (9.6  d5/2)  (3|0^Q_g  ) 

When  the  tensile  strength  of  the  concrete  has  not  been  ex- 
ceeded, f_  may  be  assumed  eq.ual  to  500  which  modifies  equation 
(11)  to 


I = 9.6  d 


- (12) 


Equation  (11)  shows  that  at  a compressive  stress  of  1500 
lb.  per  sq.  in.  in  the  concrete,  the  value  of  _!  is  only  about 
half  as  great  as  i t v/as  before  the  tensile  strength  of  the  con- 
crete was  lost.  Hence  in  analyzing  a structure  especially  for 
stresses  above  ordinary  working  stresses,  the  use  of  the  assiuip- 
tion  that  varies  throughout  directly  as  some  power  of  the  depth 
of  members  is  not  exactly  logical,  and  \iall  give  too  high  a 
value  of  at  points  of  hi^  stress.  This  is  cnnfirmed  by  test 
results  of  specimens  of  Tj^e  B and  C,  though  the  variation  is 
not  large.  For  preliminary  design,  especially  if  the  structure  is 
to  have  fairly  uniform  stresses  throu^out  the  region  of  high 
bending  moments,  it  will  usually  be  satiisfaetory  to  use  a relation 
such  as  I = kd^,  throughout  all  sections.  For  solving  statically 
indeterminate  problems  the  value  of  k is  usually  immaterial,  since 


I 


) 


i 


o 


Zi.' 


•t  ’ t> 


I/, 

■ ’JO  - ■ 


— — 85 

only  relative  values  of  ^ are  required. 

Equations  (11)  end  (12)  Ccji  not  be  expected  to  apply 
to  members  in  which  the  sliape  of  cross-section,  percentage 
of  steel  or  quality  of  concrete  vary  greatly  from  those  used 
in  these  tests.  It  is  believed,  however,  that  these  equations 
show  the  general  way  in  which  the  moment  of  inertia  varies 
in  a reiriforced  concrete  member.  Further,  the  foregoing  com- 
parison indicates  that  within  the  range  of  working  stresses  _I 
may  be  calculated  according  to  its  mathematical  definition, 
by  the  ordinary  method  of  replacing  the  area  of  steel  in  a 
section  by  an  equivalent  area  of  concrete,  or  vice  versa.  In 
either  case  the  value  of  E will  be  used  which  corresponds  to 
the  material  of  the  equivalent  section, 

14,  Deflections,-  Measurements  of  deflections  were  made 
on  each  frame,  as  explained  in  Section  8,  at  intervals  of  one 
foot  along  the  entire  frame.  Through  the  relation  vhich 

feiev 

exists  between  deflections  and  the  M diagrams  for  a 'number  in 

El 

flexure,  it  was  hoped  that  the  deflections  could  be  used  to 

study  the  variation  in  the  quantity  El,  Deflection  is  a second 

integral  function  of  the  quantity  M • Since  graphical  differoat 

El 

iation  is  not  practicable  with  any  degree  of  accuracy,  the 
exact  M diagram  corresponding  to  a given  elastic  curve  can  not 

ST 

be  found.  The  best  that  can  be  done  is  to  perform  the  reverse 
operation,  Ehov/ing  values  of  M,  and  assuming  values  of  M, 
deflection  curves  can  be  obtained  T/hich  may  be  compared  with  the 
experimental  curves. 


--irv 


V 


C'. : 0 

( ' 


':5J 


J. ,: ) r;«o^  ■^'‘:  li 


7%')  {.:.. 


orii  (C  ':t&ocr&.  ‘"'v 

t ■■  I ' 

■ ff’ .^  • 1 1 . ■•'?■  ; .1-; 


n o u :• 


SI  / 


. :zir  ^:^^eto^ICK> '^j'-’-cC'ri  ‘■‘.“■ 


'-.  ■ ■■:  o ■ 


1.  > ..  ‘f'- 


i’  ■ Ji,-*' '•■  oorfi  i>f  ■ J'/aXi*'’  ad 

,v.  .;  .rxo  ' 


m 


'♦  «■'  ■i'^ 


.10  ; ' * 


E.1-.  J 4 l>e»  : '-. 


|0>  -•  ' 

■■>  '.■  . ■ I.  ■ 

•jvi-tt 

*ju-:  -7 .'  ■-  ■ 

•“'.ir’  ,v' 

"'■■■  •*'  '."7 

• f . 

!#■ 

f)  ’•  ■ ' . .'  ■ r»f".  ‘ ' 

t • * « 

■•r'i.  :ro  ' 

"■  ' >'  L ) 1.  ' ’ 


T-*  1 lH*Ji  " ^ '''  ft*'**  I 


'^'tz  1 L 


':4-- 


» < 

.?i’  ■ '7  .:o^- (.' 


O'l..  a-‘A  »!fr:o.! .1  .';uJ.'roJj  <• 


t:'  '.‘O.-J 
; r -I  '.•  ./'  *• 


,1'' 


.1/.  ! ■ f . i'trT'-AKfl 

-yJI.  . • ' ' J 


1 

*1 


* :>vi:r 


'i.j  ' r.'T'-*>»'  *(rt>5  ''■  'y.':  rcojtCjt./ 

'.■*•'(  '•‘.  'iif' 


.'  • V i'.  e t.cc  rJji;o{j  ^ •'. ' ■ .;; 


**  f- 


f.  ':•■•  7) 


.-.(•  r •;•  Jbr  , •• 

j ’>V  ■ ■./'>  l.iV-  ■ ;!tfo 


‘ \ ,■ 

y • 


■•'7.'  a 

<*t2EWT  o. 


■]  i : 

“^7X1/0  n:.  I : ■ 


•A-'  ./  ■ , ■ 

/r  'W' 


t 'i  f . 


; ! . 


,•  A '.  ' 

."V 


l.•■^l  VJi  I * ■.i’,'*...i:i  ''i.-i'ft! 


i>’ 


V. 


86 

This  has  heen  done  for  one  loading  on  each  type  of  specimen, 
using  a constant  value  of  3,750,000  Ih.  per  sq,  ini  for  E,  and 
a value  of  1 as  obtained  from  equation  (11),  Pig*  45  to  49  show 
calculated  and  observed  deflections  graphically  for  frames  13A1, 
13B1,  13C1,13P1,  and  13E1*  The  portion  of  these  diagrams  show- 
ing calculated  deflections  have  been  constructed  by  making  use 
of  the  well-known  second  integral  relation  bet?/een  force  poly- 
gons and  funicular  polygons.  Here  the  "forces"  laid  off  in  the 
vector  diagram  to  the  right  are  the  values  of  M/l  which  have 
been  determined  from  the  known  forces  acting  on  the  frame  and 
from  equation  (11).  Choosing  the  proper  pole  distance,  which 
depends  upon  the  scales  used  in  laying  off  the  various  quantities 
a funicular  polygon  is  drawn,  starting  from  the  corner  of  the 
frame  and  making  the  string  at  point  15  horizontal.  The  funicu- 
lar polygon  represents  the  elastic  curve  for  the  specimen  and 
its  ordinates  agree  very  well  with  the  deflections  v;hich  were 
observed  during  the  test.  An  exception  is  seen  in  the  case  of 
frame  13E1,  which  differed  in  section  from  the  others,  and  to 
which  equation  (11)  does  not  ^pear  to  apply  particularly  well. 

For  use  with  any  rectangular  frame  without  brackets,  Maney’s 
equation  for  deflection*  is  readily  applicable.  For  a frame  with 
loads  at  the  one- third  points  in  which  the  maximum  moment  at  the 
center  is  Mq  = "the  maximum  deflection  at  the  center  is 

-4).  Hence  in  Maney*s  equation  f = ^^(eg+  e^) , 
the  coefficient  c becomes  — )•  The  quantities  gg  and  e^ 

must  be  measured  at  the  point  at  which  the  maximum  moment  is 

measured.  By  the  use  of  Maney*s  equation,  using  steel  deforma- 
* "Relation  between  deformation  and  deflection  in  reinforced 
concrete  beams"  by  G. A, Maney. Proceedings  A. S.T.M. -Technical 
Papers,  Vol.XIV  p.310  -1914 


I 


V ^ • 

— J 


. \ 


i 


r 


t 


\ 


■ 


87 


C0MPAR130N  OF  OBbER^ED  Am  COMPUTED  DeFLECT!ON3 
AT  EOAD  OF  4,00  0 0 POUND5 

C^P^C/MEN  • /3  A 1 


Pig.  45.  Calculated  and  Observed  Deflections,  Specimen  15A1. 


A T LOAD  OF  4,00  0 0 POUNDS 

3pf  a men  • /3  5 J 


IS 


/3 


!Z 


// 


R 


Pig.  46.  Calculated  and  Observed  Deflections,  Specimen  15B1 


88 


-/s 

-H 

/3 

-/Z 


■-^-3 

tii 


COMPAR/3CN  OF  ObFERYFD  AND  COMPUTED  DEFLECTIONS 
AT  LOAD  OF  60000  P0UND3 

Specimen -13  Cl 


Fig,  47,  Calculated  and  Observed  Deflections,  Specimen  13C1, 


AT  LOAD  OF  60000  POUND3 
Op  Ed  MEN  -/3D  I 

Fig,48,  Calculated  and  Observed  Deflections,  Specimen  15D1, 


89 


C0MPAR/30N  OF  OfFERFFD  AND  COMPUTED  DeFLECT/ONS 
AT  LOAD  OF  60000  POUND3 
3 PEC/MEN  - /3  £ I 


■IS 


■H 


t3 


■/Z 


// 


■/O 

z 


±3 

■h*7 


Pig.  49.  Calctilated  and  OlDserved  Deflections,  Specimen  13E1. 


'■'^M^y  ■■■'•>■'■  ( 4' 


'of  ',  •»  ^iV 


W.-.. 


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' ■•  -‘J  Js^V-'C  -’^^'-S'  *rs™S®*??l 

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.X^X  fremic^gP  .oaoXiotJX':^^, iJ«:^«XttoX<»CN^®  0^ 

''•/'  ■ 'i'Hi 


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p 


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90 

tions  and  using:  for  d the  actual  distance  center  to  center  of 
the  reinforcing  bars  upon  r/hich  readings  were  taken,  the 
quantities  shown  in  Table  8 were  obtained. 

These  two  comparisons  show  a very  close  agreement  between 
calculated  and  observed  readings,  and  give  further  evidence 
that  the  theoretical  relations  which  obtain  between  moments 
and  deflections  hold  true  for  these  frames. 

TABLE  8 

CALCUL^ITEI)  Am  OBSERVED  DEFLECTIOIJS  AT  MIDSPM 


Specimen  Load  k* 

lb. 

0 (eg  + eg) 

Inches 

g)  Observed 
Defl ection 
Inches 

13A1 

30 

000  .466 

.085 

.00052 

.14 

.14 

tf  tf 

60 

000  .648 

.097 

.00099 

.31 

.28 

ff  M 

90 

000  .590 

.094 

.00138 

.42 

.42 

13A2 

30 

000  .400 

.079 

.00074 

.16 

.16 

IT  n 

60 

000  .536 

.090 

.00139 

.39 

.39 

T»  n 

90 

000  .500 

.088 

.00219 

.60 

.68 

13E1 

30 

000  .314 

.067 

.00093 

.23 

.26 

FT  IT 

60 

000  .560 

.091 

.00186 

.61 

.55 

Further  use  has  been  made  of  tne  measured  deflections 
in  studying  the  variation  in  stiffness  of  each  frame  as  a 
whole  during  the  application  of  the  test  loads.  In  a homo- 
geneous beam  ivithin  the  elastic  limit  of  the  material,  the 
quantity  M is  constant  and  is  proportional  to  the  ratio  of 
load  to  deflection,  P/f,  In  these  test  specimens  the  ratio 
P/f  varied,  and  hence  the  variation  in  which  is  proportional 
to  p/f  may  be  calculated  from  measured  values  of  P/f,  On  this 
basis  Fig, 50  has  been  constructed,  using  a relative  value  of 
p/f  equal  to  unity  for  the  30  000  lb.  load  on  each  frame.  The 
value  of  f used  in  each  case  was  the  average  of  measurements 


91 

on  deflection  points  10,11,12  and  13,  near  midspan®  ¥/hile  the 
decrease  in  stiffness  ?/ith  increasing  load  indicated  in  Fig,50 
is  similar  to  that  shown  by  Fig.  43,  it  must  be  remembered  that 
in  the  former  the  deflections  are  influenced  by  the  stiffness  of 
all  sections  of  the  frame.  While  the  various  parts  of  the 
frame  are  subject  to  widely  differing  intensities  of  stress,  the 
sections  most  highly  stresses  have  the  greatest  influence  upon  the 
deflections  at  midspan.  The  decrease  in  stiffness  under  increas- 
ing load,  as  shown  by  both  deflection  and  deformation  readings, 
seems  to  be  a typical  phenomenon  of  reinforced  concrete  members. 

15,  Cone fusions. - In  analyzing  the  test  results  it  must 
be  remembered  that  the  materials  of  which  the  specimens  were  made 
are  of  rather  unusual  quality.  The  compressive  strength  and 
modulus  of  elasticity  of  the  concrete  are  much  higher  than  are 
usually  encountered  in  reinforced  concrete  construction;  in  a 
similar  way  the  steel  combined  a high  elastic  limit  with  a fairly 
high  degree  of  ductility.  Materials  of  this  quality  were  of 
especial  advantage  for  investigational  work,  but  may  not  be 
considered  as  representative  of  materials  generally  available 
for  construction  work. 

The  use  of  a large  percentage  of  longitudinal  reinforcement 
made  it  possible  to  utilize  much  of  the  compressive  strength  of 
the  concrete,  while  the  large  amount  of  web  reinforcement  used 
permitted  the  development  of  exceptionally  high  shearing  stresses 
without  diagonal  tension  failures. 

Certain  definite  effects  have  been  determined  from  the  use 
of  brackets  in  the  particular  test  pieces  described  herein,  but 


i/  \ 


'H.  * 


^ ff-,  J,>oii  "■  ■•  'c 


' . ■r"'— j r 

\ a’;! 


a. 


^ -.J  - 


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92 


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f 

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f 

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r 

/ 

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4 

□ 

< 

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/ 

A 

X 

- 

/ 

/ 

■ 

/ 

/ 

/ ^ 

/ 

• 

! 

/ 

□ 

/ 

/ 

/ 

/ 

/ 

( 

■ 

/ 

/ 

/ 

- 1 

e 

o 

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1 

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+ / 
/ 

( 

) 

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§ ^ ^ ^ ^ 

■99LUVJJ./0  e^3uj;i^Q  aboj^AD  bu/4D9ipui  j2jo93nidA  pai^di?^ 


Proportion  of  maximum  load  on  frame 

Fig,  50,  Diagram  Showing  Decrease  in  Stiffness  with  Increasing  Loads  on  Frames 


many  more  tests  are  necessary  before  any  broad  generalization 
can  be  made  concerning  the  effectiveness  which  a bracket  will 


have  in  different  types  and  shapes  of  frames*  A niimber  of 
tentative  conclusions,  ]3.owever,  may  be  formulated* 

(1)  From  an  analysis  of  the  test  data  it  appears  that  the 
reinforced  concrete  frame  can  be  treated  with  a fair  degree  of 
accuracy  by  analytical  methods  similar  to  those  used  in  arch 
analysis*  A study  has  been  made  as  to  the  validity  of  some  of 
the  assumptions  usually  employed  in  such  an  analysis. 

(2)  For  the  purpose  of  determining  the  distribution  of 
bending  moments,  it  seems  to  be  sufficiently  correct  to  consider 
the  entire  section  of  a specimen  as  effective,  even  at  points 

of  sudden  change  of  shape*  The  effect  of  such  a change  in 
shape  upon  the  stress  in  the  member  is  a matter  ?/hich  needs 
further  experimental  investigation. 

(3)  In  the  analysis  of  statically  indeterminate  frames  the 
modulus  of  elasticity  of  the  material  and  the  moment  of  inertia 
of  the  cross  section  are  quantities  of  primary  importance*  With- 
in the  range  of  working  stresses,  the  value  of  the  product  M 

as  determined  from  a large  number  of  test  readings  agrees  closely 
with  the  value  of  M as  calculated  mathematically,  using  the 
common  method  of  replacing  the  steel  area  by  an  equivalent  con- 
crete area  and  neglecting  the  tension  area  of  the  concrete  if  the 
tensile  stress  is  high.  At  higher  stresses  the  tests  indicate  a 
decrease  in  the  value  ofS,  resulting  in  a relative  loss  of 
rigidity  at  points  thus  stressed*  This  might  produce  a slight 
readjustment  of  the  moment  distribution,  and  some  leeway  should 


J 


94 

be  allowed  in  the  design  of  the  structure  to  accomodate  such 
an  occurrence.  It  is  to  be  noted,  however,  that  if  the  struc- 
ture can  be  designed  so  as  to  develop  nearly  uniform  stresses 
throughout,  there  will  be  little  variation  in  rigidity  under 
the  higher  loads, 

(4)  Fairly  consistent  q.uantitative  informatinn  as  to  the 
variation  in  M has  been  obtained  from  the  tests  of  tbe  differ- 
ent specimens.  Until  the  concrete  begins  to  fail  on  the 
tension  side  of  the  member  the  value  of  M appears  to  vary 
about  as  the  5/2  power  of  the  depth  of  the  section.  After  the 
concrete  begins  to  fail  in  tension  the  value  of  ^ gradually 
decreases,  in  the  manner  indicated  by  equation  (11), 

The  assumption  that  M can  be  expressed  by  a simple  equation, 
El  = kd^  is  evidently  not  correct,  but  will  usually  be  satis- 
factory for  preliminary  designs.  An  exponent  n equal  to  5/2 
in  the  above  expression  applied  Very  well  to  these  highly  re- 
inforced members;  with  a smaller  amount  of  reinforcement  an 
exponent  n equal  to  3 may  be  expected  to  apply,  as  it  would 
also  for  rectangular  sections  of  a homogeneous  material.  For 
determining  bending  moment  distribution  the  relative  magnitudes 
only  of  the  quantities  ^ at  different  sections  are  needed,  so 
that  the  value  of  the  coefficient  k is  immaterial  for  such 
calculations, 

(5)  Calculated  deflections  of  the  test  specimens  based  on 
values  of  moment  of  inertia  from  equation  (11)  agree  very  well 
with  measured  deflections,  and  also  with  deflections  calculated 
by  use  of  Maney*s  equation.  This  is  significant  as  showing  a 


95 

fairly  consistent  agreement  among  the  various  data  of  the 
test, 

(6)  From  calculations,  the  basis  of  ¥/hich  is  confirmed 
by  the  tests,  it  is  found  that  the  effect  of  brackets  on  the 
bending  moments  in  a frame  may  be  expressed  as  a function  of 
the  clear  span(from  edge  to  edge  of  brackets) , of  the  ratio 
of  height  to  span  of  fra.me,  and  of  the  given  loading.  The 
importance  of  the  various  factors  is  indicated  in  eq.uation  (4)  . 

(7)  The  effect  of  brackets  if  sometimes  thought  of  as  a 
shortening  of  the  span  of  the  loaded  member.  That  is,  the 
bracket  is  considered  a part  of  the  end  support  and  thus  the 
center  of  bearing  is  brought  out  from  the  center  line  of  the 
column.  It  has  been  found  that  this  shortening  of  the  span  is 
not  constant  for  a given  bracket,  but  also  varies  with  the 
ratio  of  height  to  span  of  the  frame.  For  the  frames  tested, 
the  total  span  may  be  considered  as  reduced  by  about  t^.vo-thirds 
of  the  horizontal  length  of  the  bracket  at  each  end.  While 

the  total  moment  has  been  reduced  in  this  way,  its  distribu- 
tion between  positive  and  negative  sections  has  varied.  The 
proportional  amount  of  negative  moment  increases  considerably 
as  the  size  of  bracket  increases.  Hence  while  the  decrease 
in  total  moment  is  in  effect  a shortening  of  the  span,  this 
viewpoint  does  not  lead  to  logical  conclusions,  since  the 
negative  moment  actually  increases  as  the  span  shortens. 

(8)  The  use  of  45^  brackets  in  these  tests  is  not  intended 
to  imply  that  this  shape  is  the  most  effective.  For  any  given 
frame  and  loading,  the  mdst  desirable  shape  of  bracket  may 


■i 


« • 


. 'Tv  «;\t  c 


. ■•••i  ..V  < V- 

• 

■ ’ rn-Jo 

'*V 

<*'*  t)-a5'-  t iC"  ■ , ' '•: 

* 

‘ ■ •'  ^ t-. 

,■••'•-'■  In  rs2^«i  ■ 

■ ■'■.’  '.i 

•:  L: 

V 

--  ^ ‘ '^-4 

'■;■> 

« 

. ’ .' ' f'  H ' ■ 

\ * 

' i - '■ikiJk'* 
.-*y*^55.5r  •- 

^-:o  <- '<  ■ ? • 

■•  » •.  *-■•  '■  ' • : 

*:  ‘ 1 '■  ■* 

r-.-  ,■  ..  . •■  ' '■•roo  '■ 

. f. 

^ -W’  • '0-)  ;S;.r. 


A 


<? 


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96 


iie  determined.  The  general  rale  should  he  kept  in  mind  that 
the  bending  moment  diagram  for  a frame  of  uniform  section  is 
an  approximate  influence  line  for  the  e ffectiveness  of  brackets 
or  haunches.  That  is,  an  ordinate  at  any  point  of  this  diagram 
represents  the  relative  usefulness  of  an  increase  in  section 
at  that  point. 

(9)  \^/hile  brackets  will  usually  have  rectilinear  outlines, 
it  is  evident  that  the  brackets  of  Specimen  151)1  and  13I)E,  which 
approach  a curved  form,  have  certain  advantages.  This  shape 

of  bracket  was  very  effective  in  reducing  the  moment  at  midspan, 
and  it  also  produced  a fairly  uniform  distribution  of  stress 
throughout.  The  result  was  that  these  frames  withstood  a much 
greater  load  than  any  frames  of  the  other  types. 

(10)  Brackets  should  not  be  used  for  the  purpose  of 
reducing  stresses,  without  also  determining  what  effect  they 
will  produce  upon  moment  distribution.  A large  bracket  may 
produce  a hi^  moment  at  a section  where  it  would  not  occur 
without  the  bracket  and  where  the  member  consequently  is  not 
reinforced  suff iciently.  For  example,  it  is  seen  from  the  test 
of  Specimen  131)1  that  while  the  bracket  was  deep  enough  to 
provide  for  the  large  moment  at  the  corner,  it  caused  failure 
to  occur  at  the  weaker  section  at  midheight  of  the  column. 

(11)  The  results  of  these  tests  confirm  the  theoretical 
deduction  that  brackets  can  be  used  to  effect  a considerable 
saving  of  material  and  of  dead  weight  in  a structure.  The 
bracket  eliminates  the  high  local  stress  which  is  found  at  the 
sharp  comer  of  a frame;  it  produces  a more  uniform  variation 
in  stress  along  the  frame,  thus  minimizing  the  tendency  for  the 


97 

formation  of  cracks;  and  it  reduces  bond  and  shearing  stresses 
at  the  corners  of  the  frame.  Fiirther,  by  the  careful  choice 
of  the  brackets  the  bending  moments  may  be  varied  considerably 
and  thus  a proper  balance  may  be  secured  between  the  stresses 
in  regions  of  positive  and  negative  moments. 

The  desirable  features  mentioned  apply  to  all  forms  of 
continuous  beam  and  frame  construction.  In  reinforced  concrete 
work  the  formation^  of  brackets  and  haunches  is  comparatively 
easy,  and  in  important  structures  the  saving  of  material  should 
considerably  overbalance  the  extra  cost  of  construction. 


ri’.Tw^ 

■’'W'’*  ' ■ '•  ■ --r-^vs  r ■ r t . - • T • .^...-,  •'sw.-i 

^tT2"ji.ejfr»  t ia3itiotb 


• f V -3  ^ 


'o^i 


iT?r 


Itttaf'.o  <afv‘  x^  ,'i'ySiJi^  \\esM 

/ . -■  ~'^l  * •*/',"  *’ ' ■ .c^Jj  '*  •'5' F 


S*} 


-Vi*' 


K ^»1^;*  S<f  A^TxroAti  J&jj  ooxwXjrcT.  0D^  5^Vi 

i'  ' J ■'  ''  ' ■ '^'^  '''  ''“^  ■ '*^|««pilr 

^ '‘-^  ■ ' ■ -V  .^/  ' 


«» 


XX*  of  OQC^IS'IXU^  OOT^7^gO'^:  OX<fiTl%;Vi'* 

: - /:•  '■'  ■■•■-'  ■ J-  ■ ■•  ; ■ '.^,  .-V' 


- .Q  ’*'4^ 

? .rfaiot‘vt<*Jio«^*i-o  feVo  Mfii  > »4f 


V, 


- - .1 


’V 


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vis 


4.;V-'‘-l  .:  Si4r 

^ ... 

■ <j  4 ' 1 


'V\' 


.T  V 


\,-h- 


y>^‘/S 


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fc-:; 


« V,' >ijD  '•  ■■■  ’f^'v’^'  ‘'J' ^-'’.^..t.^ 

^ ■'  ‘ /...^  ''ii. 


APPENDIX  I 


16.  Test  Data  and  Drawing.-  Detailed  data  of  all 
stressf^  deformations,  deflections,  loads  and  reactions  ob- 
served in  the  tests  are  given  in  Table  9.  While  these  data 
have  been  presented  elsewhere  in  the  form  of  curves  or 
diagrams,  they  are  tabulated  here  to  add  to  the  completeness 
of  the  thesis. 

Following  Table  9,  detailed  sketches  showing  the 
position  of  all  strain  gage  lines  and  deflection  points,  as 
well  as  the  position  of  cracks  at  failure,  are  given  in 
Fig.  51  to  67.  It  will  be  noted  that  strain  gage  points  on 
steel  are  marked  by  solid  circles,  those  on  concrete  by  open 
circles,  and  deflection  points  by  open  sq.uares.  The  gage 
lines  are  numbered  to  correspond  with  the  data  of  Table  9. 

It  is  felt  triat  the  crack  drawings  fizrnish  considerable  informa- 
tion regarding  the  behavior  of  the  frames  under  load. 

TABLE  9. 

DATA  OF  TESTS 

Note.-  In  the  table  loads  are  recorded  in  pounds, 
deflections  or  movements  in  inches,  unit  stresses  in  thousands 
of  lb.  per  sq.  in.,  and  unit  deformations  in  thousandths  of  an 
inch  per  inch.  The  + sign  indicates  tensile  stress  or  deforma- 
tion and  upward  or  outward  deflection, and  the  - sign  indicates 
the  opposite. 


SPECIIvISU 

13A1 

99. 

Load  on  Specimen  - 

lb. 

Base 

Observation 

30  000 

30  000 

60  000 

90^  bO(J" 

'120  oob 

Moved 

Outward 

End  Thmst-lb. 

6 200 

6 200 

8 200 

14  300 

17  600 

End  Movement- in. 

-.01 

0 

^.01 

—.04 

-.01 

+164 

Unit  Stress 

• 

on  G.  L.  37 

-4.5 

-3.7 

-4.5 

-6.8 

-9.8 

36 

-3.4 

-4.1 

-6.0 

-9.4 

-14.9 

35 

-10.8 

-6.4 

-6.0 

-10.1 

-18.8 

+ 9.0 

34 

+3,4 

-7.1 

-9.4 

-16.1 

-28.5 

+3.4 

30 

-4.5 

-7.5 

-7.5 

-10.5 

-7.1 

+13.9 

29 

-2.6 

-3.0 

-2.6 

-1.5 

+ 3.4 

28 

-1.5 

0 

+3.8 

+ 6.0 

+18,0 

27 

+4.9 

+ 3.8 

+8.3 

+13.9 

+ 21.4 

26 

+ 6.8 

+ 7.1 

+13.  f 

+19.9 

+ 25 . 2 

25 

+ 9.4 

+8.6 

+ 21.0 

+ 30.0 

+52.5 

24 

+ 13.5 

+14.7 

+ 23.2 

+33.0 

+52,2 

23 

+ 7.9 

+ 9.8 

+ 19.5 

+29.6 

Y.P. 

22 

+13.1 

+12.4 

+ 22.9 

+32.2 

+ 54.8 

+14.6 

21 

+ 10.1 

+13.9 

+ 21.8 

+ 31.1 

+Y.P. 

+12.8 

22a 

+ 10.1 

+ 12.4 

+ 20.6 

+ 31 . 1 

+Y.P. 

+11.3 

23a 

+ 11.6 

+ 14.2 

+ 22.5 

+ 34.1 

+Y.P. 

24a 

7.9 

+ 7.9 

+ 21.0 

+30.4 

+Y.P. 

25a 

+ 10.5 

+12.4 

+15.4 

— 

+Y.P. 

26  a 

+ 9.4 

+ 9.0 

+ 18,0 

+ 25.5 

+ 34.9 

27  a 

+ 2.6 

+5.6 

+10.9 

+18.4 

— 

28  a 

+ 5.2 

+ 6.4 

+3.0 

+8.3 

+15.0 

29a 

-1.1 

0 

-4,1 

-2.7 

+3,4 

+ 15.4 

30a 

-2.2 

-5.2 

-6.0 

-9,8 

-17.6 

+ 12.8 

34  a 

-4.9 

-4.9 

-3.0 

-13.5 

-22.1 

+5.3 

35  a 

-5.6 

+0.4 

-1.1 

-3.0 

-10.1 

+14.3 

36  a 

-5.  6 

-4.5 

-2.2 

-0.8 

-8.3 

87  a 

-3.4 

+ 6.0 

-1,8 

+ 2.3 

-0.4 

17a 

0 

+1.1 

+ 2.3 

+ 5.6 

+18.4 

16a 

+0.4 

+1.5 

+ 6.0 

+ 11.3 

+24.4 

.15a 

+4.5 

+ 6.4 

+ 10.5 

+15.4 

+ 27.0 

14  a 

+ 3.8 

+ 6.0 

+9.4 

+ 16.1 

+25.2 

-5.6 

13  a 

-f-3.8 

+4.1 

+ 7.5 

+13.5 

+30.8 

-7.1 

12a 

+7.1 

+ 9.8 

+ 12.4 

+20.3 

+47.7 

0 

10a 

+1.9 

+ 6,4 

+ 13.1 

+16.9 

+28.9 

-10.9 

9a 

+ 4.5 

+4.5 

+ 7.5 

+ 10.9 

+9.4 

-3.8 

8a 

+ 2.6 

+3.4 

+ 7.9 

+12.0 

+48 .0 

7a 

-1.9 

-1.5 

-1.1 

+4.9 

+15.0 

6a 

-1.1 

-0.4 

-2.6 

— 

+14.3 

3a 

-5.3 

-4.5 

-9.0 

-9.0 

+7.1 

2a 

-3.8 

-4.1 

-7.9 

-10.5 

-9.0 

-2.6 

1 

-4.9 

-4.1 

-7.9 

-9.4 

-2.3 

-3.4 

2 

-4.9 

-3.8 

-9.0 

-10.9 

-3.8 

-2.3 

3 

— 

-2.3 

-7.1 

-10.1 

-1.1 

6 

-3.0 

-1.9 

-4.5 

-5.6 

-3.4 

7 

0 

+1.1 

0 

+1.5 

+14.3 

it 


100 


T.-iBIE  9. 

DATA  OP  TESTS,  Continued 
SPECIMEN  13A1  Cont, 


I Load  on  Specimen  - lb.  Base 

Observation  3d000  SoUUD  606b0  96000  12OO00  Moved 

Outward 


8 

-1.1 

+0.8 

+1.1  N 

1 +3.4 

+10.9 

9 

+1.9 

+ 3.4 

+5.6 

' +8.3 



0 

10 

+ 6.5 

+ 10.1 

+15.8 

+ 24.0 

+53.6  —4.5 

11 

+ 7.5 

+10.5 

+15.8 

+23.6 

+48.4  +3.8 

12 

+3.0 

+4.9 

+8.3 

+16.9 

+36.8  -4.5 

13 

+ 5.3 

+8.6 

+11.3 

+19.5 

+56.3  -8.6 

14 

+4.1 

+ 6.4 

+16.9 

+22.9 

+34.5  +0.4 

15 

+ 3.0 

+4.9 

+10.5 

+19.5 

+33.8 

16 

+0.4 

+4.5 

+4.5 

+9.4 

+ 28.5 

17 

0 

+2.3 

+ 2.3 

+4.9 

+ 18.0 

Unit  Deform. 

on  G.L.lOl 

-.04 

+ . 04 

-.04 

-.11 

-.89 

102 

-.07 

-.06 

-.15 

-.24 

-.02 

.26 

103 

-.15 

-.15 

-.22 

-.13 

-.63 

.52 

104 

+ .07 

+ .13 

+.26 

+ .69 

— 

2.03 

105 

+ .15 

-.04 

-.02 

-.04 

— 

+ .11 

106 

+ .11 

0 

• 

1 

0 

+ .26 

+.94+1.26 

Def lec  tion-in. 

on  Point  D1 

+ .01 

-.01 

+.02 

+ . 01 

-.44 

+0.4Y 

D2 

+ .01 

0 

+ .03 

+ .04 

-.22 

+ . 32 

D3 

+01 

+ .01 

+ ,03 

+ .03 

-.02 

+ .17 

D4 

0 

+ .01 

+ .01 

+ .02 

+ .06 

+ .04 

D5 

0 

0 

0 

-.01 

-.05 

-.04 

D6 

-.03 

-.02 

-.07 

-.10 

-.43 

-.10 

D7 

-.07 

-.07 

-.13 

-.19 

-.83 

-.14 

D8 

-.09 

-.09 

-.18 

-.27 

-1.20 

-.18 

D9 

-.11 

-.12 

-.22 

-.34 

-l.EO 

-.20 

DIO 

-.13 

-.14 

-.26 

-.39 

-1.86 

-.22 

Dll 

-.15 

-.15 

-.28 

-.42 

-2.07 

-.24 

D12 

-.13 

-.14 

-.28 

-.42 

-2.13 

-.23 

D13 

-.13 

-.14 

-.26 

-.40 

-2.10 

-.23 

D14 

-.11 

-.12 

-.23 

-.36 

-1.81 

-.20 

D15 

-.09 

-.10 

-.19 

-.27 

-1.37 

-.17 

D16 

-.06 

-.07 

-.12 

-.18 

-.93 

-.13 

D17 

-.05 

-.04 

-.07 

-.10 

-.52 

-.10 

D18 

-.01 

-.01 

-.02 

-.02 

-.10 

-.04 

D19 

+ .01 

0 

+ .01 

+ .01 

+ .08 

•f*.  03 

D20 

+ .01 

+ .01 

+ .03 

+ .04 

+ .28 

+ .13 

D21 

+ .02 

+ .03 

+ .04 

+ .04 

+ .44 

+ • 25 

D22 

+ .01 

+ .01 

+ . 03 

+ .03 

+ .57 

+ . 36 

101 


TABLE  9. 

LATA  OP  TESTS,  contirmed 


SPECIIJEU  13A2 


Observation 

Load 

30000 

on  Specimen  • 
60000  90000 

- lb. 

119000 

Base 

Moved 

Outward 

End  Thrust- lb. 

^0 

10800  17500 

17>/00 

4^00 

End  Movement -in, 

• +0.02 

+0.01  +0.01 

+0.03 

+1.66 

Unit  Stress 

on  G.  L.  37 

-1.5 

-6.4  -8.3 

-10.1 

+0.4 

36 

-2.3 

-6.8  -10.5 

-18.4 

+ 1.1 

35 

-3.8 

-9.4  -15.0 

-35.3 

+ 2.6 

34 

-5.3 

-13.5  -21.0 

-44.3 

+1.5 

33 

-10.1 

-18.8  -33.0 

+Y.P. 

-0.4 

30 

-3.0 

-6.4  +1.1 

+ 22.1 

+12.4 

29 

-1.9 

-1.5  +13.5 

+13.5 

+19.5 

28 

-0.8 

+5.3  +18.0 

+ 28.9 

+13.5 

27 

+ 1.5 

+13.9  +25.9 

+33.4 

+8.3 

26 

+8.3 

+19.1  +28.5 

+35.0 

+8 . 3 

25 

+ 10.1 

+23.6  +42.4 

+70.5 

+9.8 

22 

+ 18.4 

+31.9  +46.9 

+Y.P. 

+12.3 

22a 

+14.3 

+28.1  +48.0 

+Y.P. 

+10.1 

30  a 

- 3.8 

-7.1  +0.4 

+ 21.0 

+12.8 

33a 

-7.5 

-14.2  -20.3 

—21.8 

+3.0 

^ 13a 

-*-12.4 

+14.6  +24.4 

+33.0 

-2.2 

10a 

+1.5 

+1 .5  +2.6 

+3.4 

+1.1 

2a 

-6.8 

-12.0  -19.5 

+24.0 

-6.4 

2 

-4.9 

-11.3  -17.3 

-17.3 

-4.1 

6 

-4.1 

-6.4  -5.3 

+ 9.8 

-1.5 

7 

-1.5 

-6.4  -2.3 

+ 10.2 

-1.9 

8 

-2.3 

0 +3.4 

+ 8.6 

-3.8 

9 

+1.9 

+6.4  +8.3 

+ 7.1 

-6.8 

10 

+ 7.5 

+15.0  +22.5 

+30.0 

-10.9 

11 

+ 6.0 

+16.9  +30.4 

+ Y.P. 

-3.4 

12 

+9.8 

+ 23 . 6 +42 . 8 

+ Y.P. 

-3.0 

13 

+10.5 

+22.9  +34.1 

+61.1 

-1.9 

14 

+4.9 

+12.8  +22.5 

+ 28.9 

+0.9 

15 

+ 2.3 

+8.3  +15.5 

+ 23.6 

-0.4 

16 

+ 2.3 

+4.5  +11.3 

+16.1 

+0.4 

17 

-1.9 

-0.4  +0.8 

+ 1.5 

-4.1 

IMit  Deform* n 

on  G.L.  101 

-0.07 

-0.17  -0.46 

+0 . 66 

102 

- .20 

- .53  - .85 

+ 1.22 

103 

- .30 

- .77  -1.66 

+ 0.68 

104 

0 

- .09  - .15 

-2.52 

+3.19 

105 

+ .06 

+ .46  +1.18 

+4.50 

+1.16 

106 

- .20 

- .28  ^.35 

+2.71 

Vn 


TABLE  9. 

DATA  OP  TESTS,  continued 
SPECIMEN  13A2  continued 

102. 

Load  on  Specimen 

-lb. 

Base 

Observation 

30000 

60000  90000 

119000 

Moved 

Outward 

De flection- in* 

on  Point  D1 

0 

+0.01  -0.03 

-0.84 

+0.34 

DZ 

+0.01 

0 0 

-.47 

+ .25 

D3 

+ .02 

+ * 03  + * 04 

-.10 

+ .12 

D4 

+ .01 

+.02  + .04 

+ .13 

+ .03 

D5 

-.01 

-.02  - .03 

-.03 

D6 

- .04 

-.09  - .16 

-0.51 

-.10 

D7 

- .07 

-.17  - .30 

- .98 

-.14 

D8 

- .11 

-.25  - .45 

-1.47 

-.21 

1)9 

- .13 

- . 32  - . 58 

-1.93 

-.25 

PIO 

- .15 

-.37  - .66 

-2.27 

-.27 

mi 

- .16 

-.39  - .68 

-2.47 

-.28 

m2 

- .16 

-.38  - .68 

-2.5e 

-.28 

ms 

- .14 

-.35  - .63 

-2.34 

-.26 

m4 

- .13 

-.31  - .56 

-2.00 

-.24 

ms 

- .10 

-.24  - .44 

-1.55 

-.20 

me 

- .06 

-.16  - .30 

-1.07 

-.14 

m? 

- .04 

-.09  - .17 

-.61 

-.10 

ms 

0 

-.OS  - .03 

-.13 

-.03 

D19 

+ .01 

+ * OS  + * 03 

+ .10 

+ .03 

P20 

+ .01 

+ .03  + .06 

+ .40 

+ .14 

D21 

+ .02 

+.04  + .10 

+ .68 

+ .29 

1)22 

+ .02 

+ . 04  + * 10 

+ .96 

+ .43 

TABLE  9. 

DATA  OP  TESTS,  continued 

SPECIIMT  13B1 

Load  on  Specimen-lb. 

Base 

Observation 

30000 

60000  90000  120000  152000 

Moved 

Outward 

End  Thrnst-ib* 

6750 

12600  18200 

19500  26706 

9100 

End  Movement- in. 

+•0.02 

+0.02  +0.02 

+0.06  -0.01 

+1.81 

Unit  Stress 

on  G.L.  35 

00.4 

-6.8  -1.6 

-3.2 

+8. 6 

32 

-3.4 

-10.5  -15.4 

-23.3  -28.5 

-3.8 

28 

-1.1 

-3.0  +4.5  +10.5  

+19.9 

22 

+9.0  +20.6  +30.0  +40.1  +34.9 

+15.4 

22a 

+5.3  +18.7  +29.9  +40.1  +54.4 

+13.9 

23a 

+ 7.1 

f20.3  +29.6  +40.9  +43.1 

+15.4 

24a 

+ 8.6  +19.9  -hZe.6  +3^,0  +34.5 

+14.6 

25a 

+ 6.7 

fl8.4  +29.8  +40.9  J-58.1 

+16.1 

26a 

+ 6.4  +14.2  +244  +24.4 

+17.6 

■i^'m 


TABLE  9 

DATA  OP  TESTS,  continued 
SPECII.'IEIT  13B1  continued 

103. 

Load  on  Specimen  - lb. 

Base 

Observation 

30000 

60000 

90000 

120000 

152000 

Moved 

Outward 

Unit  Stress 

on  G.  L.  27a 

+ 3,4 

+ 10.1 

+14.3 

+ 21.0 

+ 26,6 

^12.4 

28  a 

0 

+ 1.1 

+ 4.5 

+ 10.1 

+ 24.0 

+ 24.0 

51a 

+1.1 

- 4.5 

^ 9,0 

- 7.3 

-12.0 

+ 23.3 

36a 

-1.9 

- 9.0 

-17.6 

-23,3 

-22.9 

+ 3.0 

36a 

-1.5 

- 4.9 

-12.8 

-16.9 

-17.3 

+ 6.6 

37a 

-2.6 

- 5.6 

-12.4 

-14.6 

-13.9 

+ 0.4 

17a 

-1.5 

-I-  3.4 

+ 2.6 

+ 6.4 

+ 14.3 

- 3.8 

16a 

+2.6 

+ 11*3 

+18.0 

+ 18.5 

+ 26.3 

- 1.9 

15a 

+0.4 

+ 3.8 

+18.8 

+ 27.4 

+ 31  .1 

- 0.4 

14  a 

+1.5 

+10.5 

+20.3 

+ 27.8 

+27.8 

- 5.3 

13a 

+1.9 

+ 9.8 

+18.8 

+ 22.4 

+37.5 

- 3.0 

11a 

+0.8 

+ 4.5 

+ 9.8 

+ 17.6 

+20.3 

- 1.9 

10a 

+ 2,6 

+ 7.1 

-^14.5 

+20.3 

+ 20.3 

+ 2.3 

9a 

+4.1 

+ 8,3 

+14.5 

+16.1 

+22,  9 

- 2.3 

8a 

+2.6 

+ 7.5 

+ 7.9 

+11.3 

+ 8.3 

- 1,6 

7a 

+0,8 

+ 3.0 

+ 3.0 

+ 6.4 

+10.9 

+ 0.4 

6a 

+0.8 

+ 1.1 

+ 1.1 

+ 6.4 

+12 .4 

- 0.4 

2a 

+1.1 

- 0.4 

- 8.6 

- 8.3 

+ 6.4 

f 0,4 

2 

+0.8 

- 2.6 

- 5.3 

- 4.5 



.0 

9 

+1.5 

+ 3.4 

+ 6.4 

+10.3 

+ 4.9 

11 

+3.4 

+ 5.6 

+ 15.8 

+38,1 

+21.4 

+ 1.9 

14 

+4.5 

+11.3 

+18.0 

+46 . 5 

+ 27.8 

- 5.6 

Unit  Deform 

on  G.L.  101 

-0.17 

-0.39 

-0.40 

-1.47 

--  — 

— 

102 

-.13 

- ,30 

-.52 

- .99 

-1.03 

-0.24 

103 

-.18 

-.39 

-.74 

-1.05 

-.90 

+ 1.79 

104 

-.09 

-.30 

-.13 

- .20 

— 

+1.25 

105 

-.18 

-.41 

-.53 

- .81 

-.74 

-.22 

106 

-.04 

- .28 

-.15 

-.24 

+ .40 

-.18 

Deflection-in, 

on  Point  D1 

+ .02 

+ .03 

+ .03 

.04 

-.35 

+0.46 

D2 

+ .04 

+ .03 

+ .05 

.06 

-.21 

+ • 30 

D3 

+ .01 

+ . 02 

+ .04 

.06 

— 

+ .15 

D4 

-.01 

0 

0 

.01 

-.04 

+ . 02 

D5 

0 

-.01 

-.Gi 

.01 

+ .01 

-.02 

D6 

-.02 

-.04 

-.06 

.10 

— 

-.18 

D7 

-.04 

-.08 

-.15 

.25 

— 

-.23 

DB 

-.05 

-.13 

-.23 

.40 

-1.30 

-.27 

D9 

-.06 

-.16 

-.32 

.57 

-1.67 

-.32 

DIO 

-.08 

-.19 

-.37 

.66 

-1.73 

-.36 

Dll 

-.08 

-.20  • 

-.39 

.69 

-1.67 

9.36 

D12 

-.08 

-.20. 

-.39 

.68 

-1.56 

-.35 

D13 

-.08 

-.19 

-.37 

.66 

-1.53 

-.34 

« 


: 


i 


* 1 


4 

r 


TABLE  9 104 . 

LATA  OP  TESTS,  continued 
SPECBIEN  13B1,  continued 


Load  on  Specimen  -lb, Base 


Observations 

30000 

60000 

90000 

lEOOOO 

152000 

Moved 

Outward 

Deflection-in, 
on  Point  D14 

-.05 

-.15 

-.30 

-.56 

-1.21 

-.27 

D15 

-.05 

-.13 

-.24 

-.43 

—.90 

-.25 

D16 

-.04 

-.08 

-.15 

-.28 

-.56 

-.21 

D17 

T . OE 

-.04 

-.06 

— 

— 

-.14 

L18 

- 0 

0 

0 

-.01 

-.01 

-.02 

LI  9 

0 

0 

0 

0 

+ .02 

+ .01 

L20 

+ .0E 

+ .03 

+ .06 

+ . 08 

+ .20 

+ .13 

LEI 

+ ,0E 

+ .05 

+ .07 

+ .10 

+ .33 

+ .25 

LEE 

+ .03 

+ .05 

+ .07 

+.11 

+ .44 

+ .34 

TABLE  9 

LATA  OP  TESTS, continued 
SPECIMEN  13B2 

Load  on  Specimen  -lb. 

Base 

Observations 

30000  60000  90000  120000  138000 

Moved 

Outward 

End  thrust  -lb. 

5800 

1500023000 

30000 

31500 

8300 

End  Mov’t-in. 
IMit  stress 

+0.04 

+0.04  +0.05  +0.08 

+0.07 

+1.65 

on  G.L. 

35 

-2.6 

-11.3 

-15.0 

-21.4 

-55.1 

+ 7.1 

32 

-3.8 

-13.5 

-18.4 

-22.1 

-16.5 

+ 1.9 

28 

-2.3 

- 3.0 

+ 6.0 

+13.1 

-«-10.3 

+€0.6 

22 

+ 8.6 

+ 21.0 

+ 36.0 

+54.  oj 

Y.P. 

+15.0 

22a 

+ 8,6 

+19.1 

+35.6 

+5f.  C 

Y.P. 

+19.1 

25a 

+4.9 

+17.6 

+31.9 

+49,0 

+ 66. 4 

+15.0 

26a 

+2.3 

+ 13.9 

+ 26.6 

+46.5 

^58.2 

+16.9 

27a 

-0.8 

+ 0.8 

+12.0 

+20.3 

+25.5 

+10.5 

28  a 

-1.5 

+ 1.9 

+10.1 

+21.0 

+ 24.0 

+ 19.9 

31a 

-3.8 

-12,8 

-14.3 

+15.0 

-12.7 

+11.6 

32a 

-4.5 

- 7.5 

-15.4 

-24.0 

-24.0 

-1.9 

35a 

-2.3 

- 9.8 

-15.0 

-21.7 

-23.6 

+ 5.3 

36a 

-2.3 

- 8.3 

-14.6 

-2  0.6 

-20.2 

+ 3.0 

37a 

-3.4 

-7.9 

-12.4 

-17.6 

-18.8 

0 

17a 

-2.2 

1.5 

+ 5.6 

+14.6 

+13.2 

+ 0.4 

16a 

0 

+ £.8 

+16.5 

+31.5 

+30.0 

+O.4 

14a 

+1.1 

+12.4 

+ 25.5 

+39.0 

+36.8 

+ 0.4 

13a 

+1.5 

^ 7.5 

+ ES.1 

+34.1 

+34.8 

- ^.3 

11a 

+0.8 

+ 2.5 

+15.4 

+ 28.9 

+27.8 

- 0.8 

10a 

-2.6 

+ 6.0 

+12.0 

+ 13.3 

+0.8 

- 4.9 

9a 

+1.5 

+ 7.5 

+12.0 

+ 9,0 

+4.9 

- 8.6 

8a 

-0.4 

+ 3.4 

+ 3.8 

4.  3.8 

-0,4 

- 4.5 

8b 

-1.1 

+ 0.8 

+ 4.1 

+ 3.8 

+1.1 

- 3.0 

©c 

-0.4 

+ 3.4 

+ 9.8 

^10.3 

+8.6 

- 4.5 

7a 

-3.0 

- 4.1 

+ 0.4 

+ 4.9 

+4.1 

- 5.6 

i 


TABLE  9 

DATA  OP  TESTS,  continued 
SPECILCEU  13B2,  cent. 

105. 

Load  on  Specimen  - lb. 

Base 

OlDservations 

30000 

60000 

90000 

120000 

138000 

Moved 

Outward 

Unit  Stress. 

on  G,  L.  6a 

-1.9 

-1.9  . 

-0.4 

+ 1.1 

-2.6 

-1.1 

2a 

-2.6 

-8.6 

-4.1 

-9.8 

-24.0 

-2.6 

2 

-2.6 

-8.6 

-14.3 

-11.3 

-18.0 

-4.5 

9 

+1.9 

+ 9*0  + 22 . 1 

+ 31.1 

+48.4 

-1.9 

11 

0 

+9.0  +20.3 

+32.6 

+ 72.0 

-7.9 

14 

+4.5 

+14.6  +29.6 

+45.0 

Y.P. 

-6.4 

IMit  Deform. 

on  G.L.  101 

+0.03 

-0.23 

-0.40 

1 

o 

* 

o 

-1.40 

+ 6.50 

102 

- .03 

-.40 

-.55 

-.63 

-.65 

-0.30 

103 

- .45 

-.63 

-1.03 

-1.03 

-2.05 

+ 6.90 

104 

+ . 03 

+ .03 

+ . 03 

+ .68 

-.68 

+ 1.70 

105 

- .55 

-.18 

-.30 

-.80 

-.95 

-0.45 

106 

+ .70 

+ .73 

+ .73 

+1.05 

+3.60 

+ 2.25 

Deflections-in. 

on  Point  D1 

+0.01 

+0.01 

+0  * 02 

+ .03 

+0.35 

+0.43 

D2 

+ .01 

+ .02 

+ .04 

+ .07 

- -- 

+ .28 

D3 

+ .01 

4i.03 

+ .06 

+ .09 

+ .14 

D6 

-.01 

-.04 

-.07 

-.12 

-.30 

-.13 

D7 

-.03 

-.09 

-.18 

*-.30 

-.65 

-.19 

D8 

-.05 

-.14 

-.28 

-.49 

-1.00 

-.24 

D9 

-.07 

-.20 

-.39 

-.69 

-1.36 

-.20 

DIO 

-.08 

-.23 

-.45 

-.79 

-1.F8 

-.32 

Dll 

-.08 

-.24 

-.48 

-.54 

-l.f4 

-.33 

D12 

-.08 

-.24 

-.49 

-.33 

-1.80 

-.32 

D13 

-.09 

-.24 

-.47 

-.82 

-1.73 

-.32 

DD4 

-.07 

-.21 

-.42 

-.74 

-1.49 

-.30 

D15 

-.05 

-.16 

-.32 

-.57 

-1.17 

-.25 

D16 

-.03 

-.10 

-.19 

-.30 

-.74 

-.18 

D17 

-.02 

-.05 

-.09 

-.16 

-.37 

-.13 

D20 

+ .01 

+ .04 

+ .06 

+ .11 

+ .32 

+ .11 

D21 

+ .02 

+ .05 

+ .08 

+ .15 

+ .53 

+.21 

D22 

+ .03 

+ .05 

+ .08 

+ .16 

+ .72 

+ .31 

TABLE 

9 

DATA  OP  TESTS, continued 

SPECIMEN  13  ( 

>1 

Load 

on  Specimen  - 

lb. 

Base 

Observations 

30000 

60000 

90000  120000  150000  182000 

Moved 

Outward 

End  Thrust -lb. 

8200 

17000 

23600 

25000 

26700 

48000 

8200 

End  Mov't.-in. 

+ .02 

+ .02 

+ .02 

+ .02 

o 

-.03 

+1.79 

Unit  Stress 

on  G.  L.  37 

-4.9 

-9.4 

-15.4 

-22.1 

-27.7 

-56.3 

-1.9 

36 

-4.1 

-13.1 

-20.3 

-27.8 

-38.3 

r-p 

0 

33 

-3.0 

-6.4 

-10.5 

-17.3 

-15.4 

r-p 

+ 2.6 

> 


V 


J 

\ 


TABLE  9 

TEST  DATA,  continued 
SPECILIEU  15  C-1  cont. 

106. 

Load  on  Specimen 

• 

H 

1 

' Base 

Observations 

30000 

60600  POODO  12OOO0 

150000  182000 

Moved 

Outward 

Unit  Stress 

on  G.  I.  52 

-2,5 

-4.1  -6.4  -9.8 

-12.4 

-25.1 

+2,6 

51 

-5,0 

-5.4  - 6.8  -10.2 

-10,9 

-19.5 

-0,4 

50 

+1,9 

-5.0  - 6.8  -10.9 

-11.6 

+12.8 

-0.8 

27 

+1,1 

+0.8  + 2.6  + 4.9 

+ 15.1 

+41.2 

+26.7 

26 

+ 2,5 

+4,5  + 8.5  +12.8 

+ 20.5 

+45.  0 

+ 27.7 

25 

+4,1 

+9.4  +16.9  +21.0 

+ 51,8 

Vf. 

+ 25,1 

22 

+8,5 

+12  *0  +19,9  +29,5 

+41.0 

+ 68.0 

+16.5 

22a 

+ 9,0 

+15.4  +25.6  +52,6 

+42.4 

y:p. 

+ 21.4 

27a 

-2.6 

- 2,3  + 1.5  + 5,5 

+14.5 

+42.7 

+30.4 

52a 

0 

- 2.6  - 5. 6 - 7.5 

- 8.5 

-12.8 

+ 6.4 

56a 

-4.9 

-10.5  -17.7  -24.0 

-29.6 

-35.3 

+ 1.1 

15a 

+4.9 

+12.4  +17,5  +25,5 

+30.0 

+32 .5 

- 3.4 

11a 

+ 1.9 

+ 1.1  + 5.0  + 6.0 

+ 9.0 

y:r 

- 0.4 

8a 

+1.1 

+ 5.4  + 7,5  +11.3 

+ 9.0 

+ .4 

-15.9 

2a 

-2.6 

- 6.0  - 9.0  -12.8 

-15.4 

-30.4 

- 4.9 

2 

-5.0 

- 5,6  - 8,6  - 9,0 

-12.4 

-18,0 

+ 1.1 

6 

-2.6 

- 3.4  - 4.1  - 5.0 

- 3.4 

- 1.9 

-10.5 

V 

-1.5 

- 0,4  + 1.5  + 1.9 

+ 0 « 8 

0 

-11.6 

8 

+8 , 6 

+10.2  +14.6  +22.1 

+ 21.8 

+18.4 

- 4.1 

9 

+1 , 6 

0 + 5,4  + 9.0 

+ 7.5 

+10.1 

- 3.0 

10 

- 4 

+ ,4  +1.9  + 6.8 

+ 6.4 

+ 17.5 

- 2.3 

11 

+ 1.5 

+ 6.8  +12.4  +16.9 

+ 20.6 

+ 34.1 

0 

15 

+1.9 

+ 1,5  + 2.3  +15.4 

+ 22.5 

r.f? 

-..1.9 

14 

-1.1 

+3.0+  7.9  +27.8 

+40 . 8 

t?. 

- 1.5 

15 

+1.5 

+10.5  +20.6  +28.5 

+40.8 

Y.e. 

- 1.9 

16 

+1.1 

+11.6  +19.9  +30.8 

+41.6 

+ 55.0 

- 1.2 

Unit  Deform, 

on  G.  L.  101 

-.48 

-.54  -.81  -.95 

-1.27 

-1.23 

— - 

102 

-.05 

+.14  -.09  -.18 

- .16 

-.53 

+ .02 

105 

-.22 

-.48  -.54  -.95 

-1.03 

+ .46 

104 

-.05 

-.11  -.20  0 

- .56 

-.36 

+ 5.7 

105 

-.16 

-.01  -.14  -.18 

- .11 

-.66 

+ .04 

106 

-.01 

0 0 + • 24 

+ .73 

-.64 

+ .39 

Deflection-in, 

on  Point  D1 

.05 

+ .05  + ,08  + ,11 

+ .22 

— 

+ . 43 

D2 

+ ,02 

+ .04  + ,0%  + .11 

+ .21 

+ . 28 

D5 

+ .01 

+ .02  ^ .04  +.07 

+ .12 

4.  .31 

+ .16 

D6 

-.05 

-.03  ..04  -.07 

-.11 

- .30 

-.15 

D7 

-.02 

-.05  _.08  -.15 

-.22 

_ .58 

-.28 

D8 

-.04 

-.09  .,14  -.23 

-.36 

_ .89 

-.54 

D9 

-.04 

-.11  ..21  -.32 

-.49 

-1.18 

-.37 

DIO 

-.08 

-.14  ..24  -.58 

-.58 

-1.40 

-.40 

Dll 

-.08 

-.15  ..25  -.40 

-.61 

-1..51  -.41 

D12 

—08 

-.15  _.26  -.41 

-.62 

_1.55 

-.42 

D15 

-.07 

-.14  ..E4  -.39 

-.58 

-1.47 

-.40 

D14 

-.05 

-.11  -.19  -.21 

-.46 

-1.26 

-.37 

D15 

-.05 

-.09  -.15  -.24 

-.37 

- .97 

-.35 

4 


,1 


i 


107 


TABLE  9 
DATA  OP  TESTS 
SPECIMEN  13  C-1, continued 


Load 

on  Specimen  -15 

• 

Base 

Observations 

30000 

600U0 

90000 

lEOOOO 

150000 

18E000 

Moved 

Outward 

Deflection-in. 
on  Point  D16 

-.OE 

-.05 

-.08 

-.14 

-.EE 

-.63 

-.S8 

D17 

-.OE 

-.01 

-.05 

-.07 

-.11 

-.33 

-.15 

DEO 

0 

+ .0S 

+ .04 

+ .05 

+ .09 

+ .30 

+ .13 

DEI 

+ .0E 

+ .05 

+ .08 

+ .l£ 

+ .18 

+ .64 

+ .E8 

DEE 

+ .0S 

+ .05 

-«-.08 

+ .1E 

+ .S1 

+ .81 

+ .41 

TABLE  9 
DATA  OP  TESTS 
SPECIIvElT  13  C-E 


Load  on 

Specimen  - lb. 

Base 

Observations 

30000 

60000 

90000 

12000  148000 

Moved 

Outward 

End  Thrust-lb. 

11700 

16750 

E5E00 

^3500  30500 

13700 

End  Mov*t-in. 

+0.0E 

+0.03  +0.04 

+0.03  +0.03 

+1.88 

Unit  Stress 
on  G.  L.  37 

36 
33 
3E 
31 
30 

-E.3 

-3.4 

-3.7 

-1.1 

-1.1 

-3.0 

-5.6 

-7.9 

-4.1 

-3.0 

-S.3 

-4.1 

“13.1 
“17.6 
“ 9.0 
“ 7.9 
“ 6.8 
“ 6.8 

-15.0  -13.9 

-E9.1  -46.1 

-11.3  - 6.8 

- 9.8  -10.1 

-9.4  - 8.6 

- 9.8  - 7.5 

+3.8 
+ 3.0 
+ 2.3 
+1.1 
-1.1 
-9.4 

26 

+ .8 

+ 1.9  +1S.4 

EE 

-^8.3 

a.^2.8  +-22.1 

EEa 

+8.6 

+ 13.5  +24.4 

E7a 

-2.3 

+ .4  + 5*6 

32a 

-E.6 

- 5.3  “ 7.5 

36a 

—4 . 9 

-13.9 

15a 

+4.9 

+ 13.5  +21.4 

11a 

+1.1 

+ 1.5  + -^'2 

8a 

+E.6 

+ 3.8  + 

2a 

-5.6 

- 7.9  - '<^.9 

2 

-4.1 

_ 4.9  “ ^*6 

6 

-2.3 

- 3.0  “3.4 

7 

-1.9 

- .4  + 1-2 

8 

+2.6 

+7.9  + 6*0 

9 

+0.4 

+1.9  + 3.4 

10 

«0.4 

+ ^ 

11 

+E.6 

+11.3  +4.2 

13 

+ 0 

+1.5  4.5 

14 

->-1.9 

+4.9  +13.9 

15 

+2.6 

+10.1  ■^^2. 9 

16 

+E.6 

+13.5 

17 

+ 2.6 

+ 6.0  +21.4 

+ 19.5 

+40.5 

+ 19.1 

+ 33.8 

+ 57.8 

+ 18.0 

^33.8 

+ 60.4 

+ 19.1 

+ 4.9 

+ 25.9 

+ 19.1 

-IE. 4 

-12.4 

- 3.8 

-33  .0 

-60.4 

+ 3.0 

+ 31.1 

+ 21.0 

- 5.6 

+ 7.5 

+ 9.8 

- 1.1 

+ 13.1 

- 3.4 

-12.8 

-14.6 

-21.8 

- 6.0 

-12.8 

-18.4 

- 4.1 

+ 0,8 

- 4.5 

- 6.8 

+ 3.0 

- 6.0 

- 7.5 

+ 10.9 

-'3.0 

-14.3 

+ 5.3 

- 3.0 

- 7.1 

+ 4.9 

- 1.1 

- 3.4 

+ 7.5 

+ 4.-9 

- .8 

+ 15.8 

+ 17.6 

-2.3 

+31.5 

+ 31.1 

- 0.4 

+ 34.  9 

+31 .9 

- 4.1 

+49.2 

-^35'..9 

- 2.3 

+48.4 

+51..7 

- 1.1 

4' 


\ 


( 


108. 

TABLE 

9 

Data  of 

Test 

spEcncau 

13  C-2 

continued 

Load  on  Specimen  - 

lb. 

Base 

Oliservations 

30000 

60000  90000 

120000 

148000 

Moved 

Outward 

Unit  Deform 

on  Gr«D*  101 

-.35 

-.35  - 

.75 

-.67 

-.90 

— 

102 

0 

-.32  - 

.37 

-.02 

-.05 

+ .22 

103 

-.20 

-.50  - 

.75 

-.25 

-.25 

+ .63 

104 

-.10 

-.12  - 

.40 

+ .52 

-5 

— 

105 

+ .07 

-.17  - 

.40 

-.12 

-.28 

-.12 

106 

+ .05 

-.15  - 

.20 

+ .52 

+ .38 

Deflect ion- in* 

on  Point  D1 

+ .01 

+ .04  + 

.09 

+ .13 

+ .53 

+ .48 

D2 

+ .01 

+ . 04  + 

.08 

+ .16 

++40 

+ .32 

D3 

+ .01 

+ f 02  + 

.06 

+ .10 

+ .24 

+ .16 

D6 

-.01 

-.02  - 

.05 

-.08 

-.20 

-.13 

D7 

-.02 

-.04  - 

t09 

-.16 

-.40 

-.27 

D8 

-.04 

-.08  - 

.15 

-.27 

-.61 

-.32 

D9 

-.07 

-.12  - 

.24 

-.40 

-.86 

-.37 

DlU 

-.08 

-.14  - 

.27 

-.46 

-.95 

-.40 

Dll  -.09 

-.15  - 

.29 

-.48 

-1.00 

-.42 

D12 

-.09 

-.16  - 

.29 

-.49 

-l.OZ 

-.42 

D13  -.08 

-.14  - 

.27 

-.46 

-.97 

-.42 

D14 

-.07 

-.12  - 

.23 

-.41 

-.89 

-.39 

D15 

-.04 

-.08  - 

16 

-.29 

-.65 

-.36 

D16  -.03 

-.06  - 

.10 

-.18 

— 

-.32 

D17 

-.02 

-.03  - 

.06 

-r09 

-.24 

-.18 

D20 

+ .01 

+ .03  + 

+ 

.04 

+ .08 

+ .21 

+ .16 

TABLE  9 

DA.TA  OP  TESTS 

SPECIIM  13D1. 

Load 

on  Specimen$-lb. 

Base 

Observations 

300006000090000120000150000180000208500 

Moved 

End  Thrust-lb. 

90001750028400 

37300 

50000 

57300  58500 

17500 

End  Mov't.-in. 

+0.01+C 

1.03+0.04 

+0.07 

+0 .08 

+0.05  +0.06 

+ 2.04 

Unit  Stress 

- 3.8 

• 

7.5-12.4 

-16.1 

-24.0 

-34.5  -37.9 

+ 4.5 

on  G.L.  37 

' 

32 

- 4.1 

— 

4.9-  9.8 

-15.0 

-19.5 

-23.3  -23.1 

+ 3.4 

27 

0 

2.6-  1.5 

+ 0.4 

+ 3.0 

+ 4.5  +13.5 

+ 30.0 

22 

+ 6.8+10.5+18.0 

+19.5 

+ 27.8 

+33.4  +48.4 

+34.5 

22a 

+ 8.6 

-^11.6+18.4 

+ 21.0 

+ 28.5 

+34.5  +67.9 

+35.7 

25a 

+ 4.1+ 

7.1+10.9 

+ 23.3 

+ 34.5 

+40.5  +51.4 

+46.1 

26a 

+ 1.9+ 

3.8+  0.4 

+36.8 

+38.6 

+38.3  +55.9 

— 

27a 

- 0.4 

— 

0.8-  3.4 

— 4.5 

- 3.0 

- 1.1  +13.9 

+ 33.0 

28a 

- 1.9 

— 

3.8-  8.3 

- 7.9 

- 8.6 

— 9.8  + 9.0 

+ 27.8 

31a 

- 1.9 

7.1-10.9 

-14.3 

-18.0 

-24.4  -20.1 

+ 7.5 

32a 

- 1.5 

6.4-  8.6 

-12.8 

-17.3 

-22.5  -21.8 

+ 1.9 

Ji 

! 


A 


i 


110. 

TABLE  9 

BATA  OP  TESTS,  Continued. 

SPEC  lira  15B2, 


OLserv’ 

Load  on  Specimen- 

•lb. 

Base 

"^30000 

60000 

90000  120000 

150000 

180000 

210000 

232000 

ivio  V e 0. 

Out 

End 

Thrust, 

a.12000 

27500 

31500 

41700 

50000 

59200 

67100 

66300 

16800 

End 

Mov't,  +0.02 
Unit  Stress 

+0.02 

+0.02 

+0.05 

+0.06 

+0.06 

+0.07 

+0.08 

+1.74 

on  G.L. 

37 

- 5.6 

- 9.4 

- 15.4 

-22.9 

-27.8 

-36.4 

-48,4 

.X  K. 

+ 8.6 

32 

- 3.4 

- 7.1 

-10.9 

-13.9 

-21.0 

-22.9 

-27.8 

-27.7 

+ 5.3 

27 

- 2.3 

- 5.3 

- 6.0 

- 5.6 

- 5.6 

- 4.9 

- 0.4 

+16.5 

+ 27.4 

22 

+ 5.3 

+ 8.3 

+ 8.6 

+18.0 

+ 22.5 

+ 28.9 

+37.2 

+ 59 . 2 

+ 27.0 

22a 

+ 6.4 

+ 8.3 

+ 12.4 

+ 16.9 

+ 23.3 

+ 28.9 

+35.6 

+55,5 

+ 27.5 

23a 

+ 6.4 

^ 7.5 

+ 15.8 

+ 22.5 

+ 29.3 

+33.8 

+ 42.4 

y:p. 

+ 34.1 

24  a 

+ 3.8 

+ 7.9 

+ 15.8 

+ 20.3 

+ 25.5 

+31,9 

+ 39,0 

- V.p  . 

+35.3 

25a 

+ 2.3 

+ 3.4 

+ 10.5 

+ 16.9 

+ 20.7 

+ 26.6 

+34.5 

+ 59.6 

+ 28.8 

26a 

- 0.4 

- 1.5 

+ 2.6 

+17.6 

+19.9 

+ 22.5 

+30,0 

+ 63,4 

+ 64.1 

27a 

- 6.0 

- 5.6 

- 0.4 

+ 0.8 

2.3 

+ 5.3 

+12.4 

+31.5 

+ 35.6 

28a 

- 2.6 

- 6.0 

- 8.6 

- 7.1 

- 6.8 

- 5.6 

- 0.8 

+ 25,5 

+36.4 

31a 

- 4.1 

-11.3 

- 13.9 

-1^.6 

-18.0 

-19.5 

-23.3 

-23.3 

+ 9.4 

32a 

- 3.0 

- 8.6 

- 11.6 

-13.9 

-18.0 

-19.5 

-25.5 

-24.4 

+11.6 

35a 

- 4.1 

-10.9 

- 15.0 

-21.0 

-25.9 

-33.4 

-40.9 

-45.4 

+ 7.9 

36a 

-7.9 

-13.5 

- 14.6 

-20.0 

-25.1 

-33.0 

-46.1 

-51.4 

+ 4.1 

37a 

- 5.6 

-12.0 

-12.0 

-20.7 

-24.7 

-32.3 

-44.3 

-48.0 

+ 4.9 

17a 

+ 1.5 

+ 9.4 

+ 8.3 

+ 16.5 

+19.5 

+ 23.3 

+31.5 

+34.5 

- 2.6 

16a 

+ 3.0 

+13.8 

+ 9.8 

+17.3 

+ 21.8 

+ 27.0 

+39.4 

+41.3 

- 2.3 

15a 

+ 1.5 

+11.6 

+ 9.8 

+18.4 

+ 23.3 

+33,8 

+ 53.0 

•YCf. 

- 1.1 

14a 

+ 2.6 

+10.2 

+ 9.0 

+17.3 

+ 20.6 

+ 33,4 

+48,4 

+ 52.2 

- 1.1 

13a 

+ 1.5 

+ 4.1 

+ 7.1 

+ 22.1 

+36.8 

+49.2 

+ 68,3 

-Y.f. 

- 0.4 

11a 

+ 0.8 

+ 5.3 

+ 1.1 

+10.1 

+ 13.5 

+ 18.8 

+31.5 

+33.8 

- 1.9 

10a 

+ 3.0 

+ 1.9 

+ 1.1 

+ 6.8 

+ 8.6 

+10.9 

+13.9 

+13.9 

- 3.4 

9a 

+ 2.3 

+12.7 

+ 15.8 

+18.8 

+ 20.3 

+ 25.9 

+ 28.9 

+ 22.9 

0 

8a 

+ 1.1 

+ 9.8 

+ 9.4 

+15.4 

+ 19.9 

+ 24,0 

■^26.6 

+18 . 8 

- 3.8 

7a 

+ 0.8 

+ 5.3 

+ 2.3 

+ 13.5 

+ 15.4 

+ 18.8 

+20.3 

+19.9 

+ 0.4 

6a 

- 0.8 

+ 3.0 

- 2.6 

+ 4.9 

+ 8.6 

+ 12.0 

+16.1 

+15.4 

- 4.5 

2a 

-Q.4 

3.4 

- 6.4 

- 4.8 

- 4.9 

- 5.3 

- 5.6 

- 0.8 

- 2.6 

2 

- 3.0 

- 6.4 

- 9.8 

- 8.3 

- 8.3 

- 9.4 

-10.2 

- 4.9 

- 4.5 

8 

+ 1.9 

+ 6.0 

+ 6.8 

+ 21.8 

+ 24.0 

+31.4 

+37.9 

+47.6 

+ 2,3 

11 

+ 2.3 

+ 2.3 

+ 2.6 

+ 9.8 

+ 15.4 

+ 22.5 

+ 30.4 

+ 36.8 

- 3.0 

17  + 2.3  + 9.8 

Unit  Beform.on 

+ 9.4 

+19.1 

+ 26.6 

+33.0 

+42.4 

-V.K 

. 2.6 

G.L. 101 

-0.25 

-0.44 

-0.52 

-0.61 

-0.81 

-0.94 

-1.28 

-1.54 

-1.39 

102 

-0.05 

-0.24 

-0.31 

-0.44 

-0.59 

-0.66 

-0.72 

-0.93 

+0.04 

103 

-0.18 

-0.35 

-0.37 

-0.70 

-0.81 

-1.18 

-1.18 

-1.18 

+0.13 

104 

-0.12 

-0.20 

-0.17 

-0.40 

-0.40 

-0.54 

-0.63 

-0.81 

+0.15 

105 

-0.20 

-0.20-0.26 

-0.40 

-0.37 

-0.31 

-0.35 

-0.72 

+0.04 

106 

-0.10 

-0.15 

-0.18 

-0.26 

-0.24 

-0.28 

-0.24 

-0.40 

+0.09 

TABLE  9 

DATA  OF  TESTS,  Continued . 
SPECIHEE  13D2,Cont. 

111. 

Load 

on  Specimen- 

lb. 

Base 

Observ’ 

n 

. Moved 

30000 

60000 

90000  120000 

150000 

180000 

210000 

232000 

Out 

Deflection  at 

pt.Dl 

+ .02 

+ . 04 

+ .04 

+ .05 

+ .08 

+ .10 

+ .14 

— 

+ .38 

D2 

+ .01 

+ .03 

+ .04 

+ .05 

+ .08 

+ .11 

+ .14 

+ .45 

+ .26 

D3 

+ .01 

^.02 

+ .03 

+ . 04 

+ .06 

+ .08 

+ .10 

+ .27 

+ .14 

D6 

-.01 

-.01 

-.02 

-.03 

-.05 

-.06 

-.09 

-.21 

-.12 

D7 

-.03 

-.05 

-.06 

-.09 

-.12 

-.16 

-.21 

-.46 

-.25 

D8 

-.03 

-.08 

-.09 

-.15 

-.21 

-.27 

-.36 

— 

-.35 

D9 

-.05 

-.12 

-.14 

-.23 

-.30 

-.39 

-.53 

-1.05 

-.42 

DIO 

-.07 

-.15 

-.18 

-.28 

-.36 

-.47 

-.62 

-1.22 

-.47 

Dll 

-.06 

-.15 

-.18 

-.29 

-.38 

-.50 

-.66 

-1.32 

-.47 

D12 

-.07 

-.16 

-.19 

-.30 

-.39 

-.51 

-.68 

-1.36 

-.50 

D13 

-.05 

-.15 

-.18 

-.29 

-.38 

-.47 

-.65 

-1.33 

-.48 

D14 

-.05 

-.13 

-.15 

-.25 

-.34 

-.44 

-.58 

-1.18 

-.44 

D15 

-.03 

-.08 

-.10 

-.17 

-.23 

-.30 

-.41 

-.88 

-.37 

D16 

-.03 

-.06 

-.07 

-.12 

-.15 

-.19 

-.26 

-.57 

-.29 

D17 

-.02 

-.04 

-.04 

-.06 

-.07 

-.10 

-.13 

-.30 

-.16 

D20 

+ .01 

+ .03 

+ .03 

+ .06 

+ .07 

+ .08 

+ .11 

+ .23 

+ .14 

D21 

+ .02 

+ .04 

+ .06 

+ .08 

+ .10 

+ .13 

+ .18 

+ .43 

+ .27 

D22 

+ .02 

+ .05 

+ .06 

+ .09 

+ .12 

+ .15 

+ .20 

+ .57 

+ .40 

DATA  OF 

TESTS 

SPECIMEN 

13E1 

Load 

on  Specimen 

- lb. 

1 • 

UDservaxion 

30000 

60000 

77000 

End  Thrust  ,1b. 

8000 

10100 

14500 

End  Movement, in. 

0.06 

0.05 

0.05 

Unit  Stress  on 

Gage  Line  37 

- 

1.9 

- 4.9 

- 6,4 

36 

— 

4.9 

-.9.0 

-11.3 

35 

— 

6.0 

-12.0 

-14.1 

34 

- 

7.9 

-15.0 

-18.0 

33 

- 

9.0 

-16.1 

-19,1 

30 

- 

4.1 

- 7.1 

- 7.9 

29 

— 

3.8 

- 3.4 

+ 2.6 

28 

— 

2.3 

+ 0.7 

+ 6.8 

27 

+ 

4.1 

+12.4 

+19.5 

26 

4- 

9.0 

+ 22.5 

+ 29.2 

25 

+ 15.0 

+ 32.6 

+42.8 

22 

+ 17.2 

+32.3 

+43.6 

22a 

+19.1 

+ 37.5 

+ 52.1 

118. 

TABLE  9 

BATA  OP  TESTS,  Continued 
SPECniEil  15E1,  Gont. 


OLserv'n 

Load  on  Specimen- lb. 

on 

Specimen 

. 

H 

1 

30000 

60000 

77000 

30000 

60000 

77000 

Unit  Stress  on 

Beflection 

at 

G,  L . 

Point 

30a 

- 9.0 

-12.8 

- 1.5 

B1 

+ .04 

+ .09 

+ .31 

33a 

- 8.6 

-16.1 

-12.4 

B2 

+ .05 

+ .09 

+ . 26 

13a 

+ 10.5 

+19.1 

+21 . 8 

B3 

+ .03 

+ .07 

+ .18 

10a 

+ 9.4 

+ 19.1 

+ 25.4 

B4 

+ .02 

+ .04 

+ .10 

2a 

9.4 

-21.0 

-11.6 

B5 

-.01 

-.03 

-.08 

2 

-10.1 

-21.0 

+ 9.0 

B6 

-.07 

+ .14 

-.31 

6 

- 4.5 

- 9.0 

- 1.5 

B7 

-.12 

-.25 

-.55 

7 

+ 3.4 

+ 4.5 

+ 12.4 

B8 

-.17 

-.35 

-.79 

8 

+ 1.9 

+ 4.1 

+ 9.0 

B9 

-.21 

-.45 

-1.02 

9 

+ 4.5 

+ 9.0 

+ 10.9 

BIO 

-.25 

-.52 

-1.15 

10 

+ 6.4 

+14.2 

+16.5 

Bll 

-.26 

-.55 

-1.36 

11 

+ 9.0 

+ 17.6 

+ 24.8 

B12 

-.26 

-.55 

-1.46 

12 

+ 7.1 

+15.0 

+ 13.9 

B13 

-.24 

-.52 

-1.43 

13 

+ 8.2 

+ 16.9 

+ 19.5 

B14 

-.22 

-.46 

-1.23 

14 

+ 5.6 

+ 12.0 

+ 14.6 

B15 

-.17 

-.35 

— .88 

15 

+ 6.0 

+12.4 

+ 15.4 

B16 

-.12 

-.25 

-.57 

16 

+ 1.5 

+ 6.8 

+10.1 

B17 

-.06 

-.13 

-.27 

17 

+ 3.9 

+ 5.3 

^+  6.4 

B18 

-.01 

-.02 

-.01 

B19 

+ .02 

+ .02 

+ .01 

Unit  Beformation 

B20 

+ .03 

+ .04 

-.03 

on  (x.  L. 

B21 

+ .06 

-.06 

-.07 

101 

-0.22 

-0.37 

-0.64 

B22 

+ .07 

+ .06 

-.14 

102 

-0i44 

-0.92 

-1.60 

103 

-0.28 

-0.39 

-0.53 

104 

-0.05 

-0.13 

-0.09 

105 

-0.02 

+0.97 

+2.39 

106 

-0.09 

+0.02 

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Fig,  57,  Position  of  Gage  Lines  and  Cracks,  Specimen  13B2 


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Pig.  65 o Position  of  Gage  Lines  and  Cracks,  Specimen  131)2 


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APPENDIX  II 


li^l, 

17.  Supplementary  Tests  of  Paper  Models.-  In  February, 
1922,  a method  was  described  by  Prof.  G.  E.  Beggs*  whereby 
stresses  in  indeterminate  structures  might  be  determined  by 
measuring  certain  deflections  in  a paper  or  cardboard  model  of 
the  structure.  It  was  evident  that  the  method,  if  practicable, 
would  be  extremely  useful  in  connection  with  the  problems  of 
this  thesis.  Accordingly,  through  the  assistance  of  lAr.R.  L. 
BroYm  of  the  Engineering  Experiment  Station  Staff,  in  the 
limited  time  available  since  the  method  was  published,  the 
effects  of  several  types  of  brackets  have  been  measured  by 
the  use  of  this  method. 

The  theory  used  in  the  tests  of  models  is  not  new,  being 
based  upon  Max?;ell’s  v/ell-knovm  "Theorem  of  Reciprocal  Dis- 
placements". The  novel  feature  is  the  use  of  small  models  of 
paper  or  other  isotropic  material  and  the  measurement  of  de- 
flections by  means  of  microscopes  and  micrometer  gages.  The 
theory  and  procedure  in  the  tests  may  be  described  with  reference 
to  Pig. 69,  which  shows  the  arrangement  of  apparatus  which  was 
utilized  for  the  purpose.  The  model  of  the  frame  to  be  tested  is 
placed  on  a horizontal  surface  and  supported  on  ball  bearings  to 
reduce  friction.  The  hinge  B is  held  stationary  by  means  of  a 
needle,  while  the  hinge  A is  attached  by  means  of  another  needle 
to  the  screw  micrometer  D.  The  formation  of  the  hinge  between 
paper  and  needle  is  difficult,  as  the  fit  must  not  be  tight 

^ "An  Accurate  Mechanical  Solution  of  Statically  indeterminate 
Structures  by  Use  of  Paper  Models  and  Special  Gages",  by  G.E. 
Beggs.  Proc.  American  Concrete  Institute,  1922. 


152 


enough  to  cause  high  frictional  resistance  to  turning  and  it 

must  not  be  loose  enough  to  allow  play  in  the  hinge*  Assuming 

that  it  is  desired  to  find  the  horizontal  reaction  H at  A and 

B due  to  a load  P acting  at  the  point  C,  the  procedure  is  to 

move  the  hinge  A to  the  ri^t  a distance  d by  means  of  the 

a 

micrometer  I)  and  with  the  microscope  to  read  the  movement  do 

of  the  point  C in  the  direction  of  the  load  P (which  is  taken 

in  this  case  to  be  at  right  angles  to  the  direction  BA)*  By 

^c  H 

the  application  of  Maxwell’s  theorem  the  ratio  = p ♦ or  the 
horizontal  reaction  H = P 


In  practice  it  was  found 


"a 


advisable  to  repeat  the  operation,  moving  the  hinge  A to  the 
left  of  its  initial  position  and  measuring  d^,  and  using  the 
numerical  average  of  several  sets  of  observations  in  calculating 
the  value  of  H* 

The  paper  models  tested  were  of  Types  A,B,C,and  B,  as 
described  in  Section  6, except  that  eight  different  heights  of 
each  type  of  frame  were  used*  To  eliminate  differences  in  the 
q,uality  of  the  paper  of  the  models,  all  were  made  from  one 
sheet  of  paper,  the  frame  of  Type  D being  first  cut  out  and 
tested  without  damaging  it,  the  brackets  then  cut  down  to  the 
form  of  Tjrpe  C and  later  of  Types  B and  A*  The  paper  showed 
some  variability  in  stiffness  and  the  values  recorded  were  the 
average  of  a number  of  observations*  The  values  of  H/P, 
the  ratio  of  horizontal  reaction  to  vertical  load  applied  at 
the  1/3  points  of  the  top  member  of  the  bent,  are  plotted  in 
Pig* 70*  For  comparison,  values  of  H/P  have  been  calculated 
from  equation  5,  assuming  the  haunch  of  Type  I)  to  be  equiva- 
lent to  a 45®  bracket  4/3  as  long  as  that  of  Type  C, 


133 


toid  are  plotted  in  Fig,  71,  The  values  from  the  model  tests  and 
from  the  calculations  are  also  plotted  for  the  four  types  of 
frame  in  Fig,  72  to  75,  inclusive.  The  agreement  in  the  results 
from  the  two  methods  is  very  close  and  may  he  considered  a 
satisfactory  verification  of  equations  4 and  5, 

Another  use  has  been  made  of  the  models  in  determining 
relative  deflections  at  various  points  on  the  top  member  of 
the  frame.  Fig,  76  shows  relative  deflections  of  the  top 
members  of  the  models  of  Types  A,  B and  C,  These  curves  are 
in  effect  influence  lines*  for  the  horizontal  reactions  of 
the  frame  with  a vertical  load  on  the  top  member.  It  should 
be  noted,  however,  that  the  purpose  of  these  curves  is  to 
compare  values  of  H/P  with  loads  at  different  points  on  the 
same  frame  and  not  to  compare  values  bet>.veen  different  types 
of  frame.  It  is  found  that  the  horizontal  reactions  with  loads 
at  midspan  and  at  the  l/3  points  are  in  the  ratios  of  1.15, 

1.16,  and  1.17,  for  Types  A,  B and  C,  respectively;  with  uni- 
form loads  and  l/3  point  loads  the  reactions  are  in  the 
ratios  0.75,  0.73  and  0.71  for  the  frames,  respectively.  Com- 
paring these  ratios  with  those  calculated  in  Section  4,  where 
for  Type  A the  ratio  bet^;/een  the  reactions  for  loads  at  midspan 
and  1/3  points  ivas  taken  at  1.125  and  the  ratio  for  uniform 
load  and  l/3  point  load  was  taken  at  0.75,  the  agreement  is 
seen  to  be  very  close  and  the  correctness  of  the  basis  of 
equations  6 and  7 is  thus  verified. 

In  conclusion  it  may  be  said  that  the  tests  on  paper 
models  gave  results  which  were  remarkably  close  to  those  found 

* See  footnote  on  page  19, 


154 


by  analysis.  However,  individual  observations  varied  consider- 
ably from  131  e mean  value  found  and  it  was  necessary  to  take 
quite  a number  of  readings  to  eliminate  errors  of  observation 
and  manipulation.  Further,  preliminary  tests  ?7ith  these 
models  indicated  that  some  grades  of  paper  are  not  isotropic 
or  are  not  uniform  in  certain  properties,  so  that  great  care 
must  be  used  in  the  selection  of  the  material  if  such  tests 
are  to  be  used  for  scientific  work. 


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Fig.  7!  Va/ues  of  H/p  from  CalculaHom,  u^ing  FguaFon  5. 


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